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Handbook of Induction Heating

Valery Rudnev, Don Loveless, Raymond L. Cook

Theoretical Background

Publication details https://www.routledgehandbooks.com/doi/10.1201/9781315117485-3 Valery Rudnev, Don Loveless, Raymond L. Cook Published online on: 11 Jul 2017

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Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 and some other related some other aspects. and phenomena, heat simulations, and process transfer electromagnetic on the focuses chapter ChapterThis in reviewed are 4. processing inductionthermal of specifics microstructural fieldintensity, Metallurgical/ depend on magnetic als microstructure. and temperature, physical the of because materi properties nonlinear highly and interrelated tightly are that analysis phenomena, heat circuit and of transfer,tion electromagnetic, metallurgical amultiphysical is (IH) Induction heating phenomenon acomplex comprising interac BackgroundTheoretical 3 These currents produce heat currents ( Joule by the These effect however, current; coil opposite the is current. coil as to the frequency direction same their have induced the currents coil. These located the conductive are near that trically objects elec other induced be in workpiece also the in will coil. Eddy located the inside currents fieldinduces eddy magnetic currents This current. coil the as frequency same the has that (AC) 3.1) coil, current Figure multiturn alternating circuit. an the coil flow in result in will college physics. voltageincluding alternating An applied (e.g., coil induction to an solenoid several textbooks in phenomena of have IH basic discussed The electromagnetic been 3.1 phenomena, to but not including limited electromagnetic eral sev with associated is workpiece. distribution the in gradients current Nonuniform ture tempera causes heat nonuniformity source inductor This workpiece and not is uniform. chapter. this in workpiece. used be approach will acylindrical This surrounds solenoid-type that coil it convenient is inductor types, to basic review aconventional principles of considering IH of variety endless an almost is there that IH. Recognizing in of used inductor geometries An alternating coil current produces in its surroundings a time-variable magnetic field magnetic atime-variable produces its surroundings in current coil alternating An Because of several electromagnetic phenomena, the current distribution within an an within distribution phenomena, current of the severalBecause electromagnetic 2. 4. 3. 5. 1. Basic Electromagnetic Phenomena IH in Basic Electromagnetic End and edge effects, and some others edge and End and effects, Slot effect effect Ring effect Proximity effect Skin I 2 R ). 3.2 Figure shows asample of avariety 51 - - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 52 company.) Group Inductotherm Inc., an of Inductoheat (Courtesy IH. in used geometries of inductor of avariety Sample FIGURE 3.2 inductor. solenoid multiturn Conventional FIGURE 3.1 Handbook of Induction Heating ofInduction Handbook Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 of material/phase alattice in achange is transformation) (unlessthere temperature of temperature. afunction as materials of used some commonly resistivities electrical For size. ρ most metals, grain and microstructure, Ni-based superalloys). steel, tungsten, steel, stainless carbon copper, alloys (e.g., and gold, metals high-resistive and aluminum) magnesium, titanium, low-resistive considered being are alloys that and (e.g., metals metals are There ties. silver, resistivi electrical of divided their on several magnitude subgroups on the also based conductors, turn, to electrical are, be they in known are materials Although most metallic tics, etc.). Table 3.1 shows values at of [66]. ρfor temperature room materials common flow materials compared current toto other the (e.g., resistance electrical ceramics, plas text. this in used primarily is resistivity electrical practice; however, of of data majority consist data for books the mho/m ρare σ and Ω and ity ity conductiv by electrical specified is current conduct to easily of material ability electric The 3.1.1.1 systems. IH the of have that on performance most properties pronounced the effect on those only text this properties, we concentrate in electromagnetic of all importance the Recognizing others. permeability, dipole susceptibility, permittivity, moment, magnetic magnetic many and permeability,tive magnetic flux saturation density, coercive force, initial hysteresis loss, conductivity), (electrical resistivity electrical rela including characteristics of engineering quite abroad is to refers anumber expression that of materials properties Electromagnetic 3.1.1 materials. rent flow, of heated properties of imperativeelectromagnetic nature theis it understand to 18–21,44].andeddy cur field distribution amagnetic factors affecting Before the exploring Background Theoretical where typically sharply increased (Figure 3.5). (Figure increased sharply typically is of metals resistivity of temperature. At point, electrical melting the afunction grades as example,versus temperature. an As 3.4 Figure shows graphite ρof one commercial of the of ρ function graphite, to anonlinear of owing etc.), temperature function anonlinear αis value anegative.the be of αcan steels, (including alloyed carbon For steels, materials other 1/°C. Table [67]. values of 3.2 metals the of pure consists αfor selected perature perature The resistivity of the pure metals can often be approximated as a linear function of the of the function approximated be often alinear as can metals pure of the resistivity The composition, temperature, with chemical varies materials of metallic resistivity Electrical conductors have and considered electrical to alloys good are be and much less Metals These effects play an important role in the performance of an IH system IH [1,2,5–13, of an performance the role play in important effects an These decreases with temperature and, therefore, temperature conductive with For ρdecreases some materials, electrically is electrical resistivity ρ resistivity σ [45–49]. electrical conductivity of σis the reciprocal The Electromagnetic Properties of Materials Metallic Properties Electromagnetic

ρ Electrical Resistivity (Electrical Conductivity) (Electrical Resistivity Electrical 0 is the electrical resistivity at T ambient temperature resistivity electrical the is T is the temperature coefficient of the electrical resistivity. unit for αis The the coefficient electrical temperature of the αis , and *m, respectively. Both characteristics can be used in engineering engineering in used be *m, can respectively. characteristics Both ρρ () TT =+ 00 [( 1 α rises with temperature. Figure 3.3 temperature. Figure shows with rises − T )], (3.1) 0 , ρ ( T ρ ) is the resistivity at tem resistivity the ) is . Therefore, value the of . The SI units for SI units . The 53 ------​ Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 as expressed be can parameters these between relationship The circuit. electrical of an aproperty is which but to a smaller extent compared to the effect of temperature and chemical composition. chemical of and temperature but extent effect compared to to the asmaller grains), to finer corresponds deformation,plastic treatment,heatandsome factors, other of density. increase an with ρdecreases materials, content. carbon the For in powder increase an metallurgy with increases steels carbon concentration of the of plain with alloys. For resistivity example,increases electrical the values of ρ average the using resistivity. assumption physical Incorrect electrical properties, including phenomenon for Cu–Ni this trates alloys. to 50%at elements concentration weight of equal the the of alloying [47]. 3.7a Figure illus resistivity electrical maximum has often curve This curve. by represented abell-shaped is [46]. of iron resistivity residual and elements on electrical most alloying common behavior of ρto aconsiderable 3.6 Figure of illustration, the shows extent. an effect As an 54 One should not confuse electrical resistivity ρ( resistivity electrical shouldOne not confuse (e.g., size grain by the affected also ρtypically is higher resistivity valueThe of electrical or decreases ρcontinuously curve, bell-shaped the of having instead some cases, In of the variations regarding alloys, IH to have itWhen understanding important is aclear alloys, behavior of elementsFor the concentration ρversus the of some binary alloying the affect can and lattice metal the distort alloying and metals in observed Impurities can result in costly misleading postulation. costly misleading in result can Source: Teflon Mica Glass Wood (dry) Graphite Nichrome Titanium Lead Stainless steel Mild carbonsteel Cobalt Nickel Zinc Tungsten Aluminum Gold Copper Silver Temperature) (at Room Material Materials Common for Some Resistivities Electrical TABLE 3.1 Temperatures, Metallurgia, Moscow, 1989. V. Zinoviev, ThermalProperties ofMetalsatHigh R = ρ a l ,

Electrical Resistivity Resistivity Electrical Ω *m) with electrical resistance R resistance *m) electrical with 10 10 10 ( μ >10 0.42 0.21 0.16 0.09 0.068 0.059 0.054 0.027 0.024 0.017 0.015 17 16 14 7–9 Ω 0.7 –10 –10 –10 1 *m) 19 Handbook of Induction Heating ofInduction Handbook 21 20 17 (3.2) ( Ω ), ), - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 Relative magnetic permeability μ Relative permeability magnetic 3.1.1.2 appropriate the in text the sections. in further discussed system induction of is parameter the of ρon aparticular effect An others. and impedance, of depth heat efficiency, including tem generation,heating distribution, temperature coil Background Theoretical cross section where the current is flowing through. flowing is where current the section cross where Both μ Both a air. vacuum or than field to conduct ofbetter a material electric ability the the ε indicates a air. vacuum or than constant)Relative dielectric (or fluxbetter permittivity magnetic the Electrical resistivities of some commercially used metals. used commercially of some resistivities Electrical FIGURE 3.3 Electrical resistivity affects practically all important parameters of an induction sys induction of an parameters important all practically affects resistivity Electrical r

is the area of the conductor’s of conductor the area the ais and current-carrying of the length the l is are nondimensional parameters and have very similar meanings. have and parameters very similar nondimensional εare and Magnetic Permeability and Relative Permittivity (Dielectric Constant) (Dielectric Relative Permittivity and Permeability Magnetic

Electrical resistivity, µΩ*mEle0.02 ctrical0.04 resistivit0.06 0.08 y, µΩ*m 0.12 Electrical resistivity, µΩ*m 0.1 0.2 0.3 0.4 0.5 0.1 0.4 0.8 1.2 1.6 0 0 0 0 0 0 100 500 r 200 indicates the ability of a material (e.g., of amaterial ability the indicates metal) to conduct 200 1000 300 Te Te Te 400 mp mp mp 400 erature, °C 1500 erature, °C erature, °C 500 600 T Ti St Carbon s ungston ainless ste tanium 2000 600 700 te Magn Silver Aluminum Copp 800 el 2500 el Lead Ti Zi Nickel er 800 esiu n nc m 1000 300 900 0 55 - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 56 There is noticeable increase in electrical resistivity of metals near the melting point. melting the near of metals resistivity electrical in increase noticeable is There FIGURE 3.5 (ATL) of graphite temperature. versus resistivity Electrical FIGURE 3.4

Electrical resistivity (µΩ*m) 9.2 7.2 8.2 Electrical resistivity 6.2 0 Source: Zinc Tungsten Titanium Silver Nickel Nichrome Lead Iron Gold Copper Cobalt Aluminum Temperature) (at Room Metals TemperatureSome forMetals Coefficient TABLE 3.2 200 Moscow, 1991. tors), andE.Meilikhov(edi- I. Grigor’ev 400 Melting Physical Values, Energoatomizdat, Te 600 Te mp mp po erature (°C eratur in 800 t e 1000 ) 1200 α (1/°C) 0.0042 0.0045 0.0035 0.004 0.0069 0.0004 0.0037 0.005 0.0035 0.004 0.0053 0.0043 Handbook of Induction Heating ofInduction Handbook 1400 1600 Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3

(the vacuum), ε constant the similarly and ing applications. ing but materials, measurable it impact when plays metallic IH heat a major dielectric role in have not does usually a Relative effects. permittivity ring and proximity well as as effect, edge, end and electromagnetic effect, skin phenomena, the including electrical affecting Background Theoretical free space. free IEEE Press, 1993.) Press, IEEE Ferromagnetism Bozorth, R. from (Recreated iron. in element of alloying percentage versus resistivity Electrical FIGURE 3.6 C. Walton, T. Handbook Opar, Castings Iron from (Recreated of steels. those to relative iron ductile and of gray conductivities thermal and electrical the Alloys Metallic Binary of Resistivities Electrical of Handbook CRC Schroder, K. from (Recreated temperatures. different at two alloys of Cu–Ni binary resistivity (a) Electrical FIGURE 3.7 of the magnetic flux density ( flux density magnetic of the (a) The constant μ constant The selection parameters on process effect amarked has Relative permeability magnetic The productThe of μ

0.1 Electrical0.2 resistivity0.3 ,0.4 μΩ*m0.5 0.6 0 Cu 02 Pe rcent 04 age ofNicom 500°C (932°F) 0 0°C (32°F =4 r 06 and μ and

× 10 π × Electrical resistivity (µΩ*m) 0.16 0.18 0.22 0.12 0.14 0.1 0.2 ) 08 0 0 and corresponds to the ratio to the corresponds μand permeability magnetic called is po −7 nent H/m [or Wb/(A.m)] space of free permeability the called is B ) to magnetic field intensity ( fieldintensity ) to magnetic 0 Pe , % H B 0.4 rcent , Iron Castings Society, Inc., 1981.) Society, Castings , Iron Ni == 100 µµ age ofalloyingelemen rr 0.8 00 0 =8.854 ×10 or (b Al , CRC Press, Inc.,, CRC 1983.) on Press, shape of graphite (b) Effect ) Electrical and thermal conductivities 1.2 BH of cast irons relative to steel 0.5 1.5 2.5 0 1 2 3 µµ ts 1.6 −12 iniro Cu F/m of permittivity the called is D Sphe

gr uctile iron shap H n aphite ). Relative ele rica 2 e Relative therma l conductivity conductivity W Mn Mo Si ctrica l l Gray gr shap Flake aphite iron St e ee l (3.3) 57 - , Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 less than 1( than less netic steels commonly used in IH can vary from small values (e.g., small from vary can IH in used commonly steels netic μ ferences in the intensity of the magnetic field magnetic of the intensity power).(coil the in ferences For example,the μ ( related to magnetic susceptibility ( susceptibility related to magnetic closely is simply permeability. relative permeability magnetic permeability The magnetic physics here we and simply textbooks texts. refer to those 58 Effect of temperature and fi and of temperature Effect FIGURE 3.8 temperature and frequency can have can anoticeably value frequency and temperature of different μ 3.8a, grade steel atFigure carbon same from same the see the temperature. one can As composition,ical size, frequency, prior fieldintensity, treatment, grain magnetic and steels that belong to the group of ferromagnetic materials. belong group that of to the steels ferromagnetic of number alloy alarge also are There ferromagnetic. are steels carbon plain perature. All at below temperatures tem room ferromagnetic are metals earth Some rare nickel. and include at cobalt, iron, properties temperature. room These ferromagnetic revealthat the the high value of relative magnetic permeability ( value high ofthe relative permeability magnetic (e.g.,materials copper, tungsten). and aluminum, titanium, simply nonmagnetic to are as practice, referred IH materials in materials, those magnetic plain carbon steel versus carbon content carbon plain (b). In other words, magnetic susceptibility shows μ by amount which the words, other susceptibility In magnetic als is slightly greater than 1( slightly than is greater als materi of paramagnetic Relative permeability ferromagnetic. and magnetic diamagnetic,

and H B and Basic definitions, interrelations between these two properties of the magnetic fieldmagnetic thetwo of properties these between interrelations Basic definitions, In everyday engineering language, the IH professionals often call the relative the magnetic call often professionals IH language, the everydayIn engineering The ferromagnetic property of the material is a complex function of structure, chem of a complex structure, is material of the function property ferromagnetic The In contrast to paramagnetic and diamagnetic materials, ferromagnetic materials exhibit exhibit materials ferromagnetic materials, diamagnetic and contrast to paramagnetic In All materials based on their magnetization ability can be divided into paramagnetic, divided be paramagnetic, into can ability magnetization on their based materials All (a) Relative magnetic permeability 200 100 120 140 160 180 20 40 60 80 1 0 ), and its designations are discussed in high school science textbooks and college and textbooks science school high in ), discussed are its designations and μ 100 r < 1). Because of insignificant differences of μ <1).differences of insignificant Because Induction tempering, Induction hardening Hot andwarm stress relieving 200 forming eld intensity on relative magnetic permeability μ permeability magnetic relative on intensity eld 300 Temperature, °C 400 μ r >1). value The of μ µχ rr 500 χ =+ ) by the expression ) by the 600 11 Curie point or 7008 χµ =− μ r r 00 >> 1). afew only elements are There for diamagnetic materials is slightly is materials for diamagnetic . (b) r (3.4) Temperature, °C dia and paramagnetic for both 680 720 760 800 840 880 920 Handbook of Induction Heating ofInduction Handbook 0 r (a) and the Curie temperature of (a) temperature Curie the and M G Carbon content(wt.%) Curie point 0.4 (A O r differs from unity. from differs 2 ) r =2or 3) to very 0.8 S r owing to dif owing 1.2 r of mag of 1.6 ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 H density density 0.1% content. carbon as as (Curie Curie temperature the to referred called point) often is also and nonmagnetic ing temperature. values (e.g.,high H field 500),intensity magnetic more on the than depending Background Theoretical The magnetic field intensity fieldintensity magnetic The (a) Magnetic field intensity ( intensity (a) field Magnetic FIGURE 3.9 the assumption of using an electromagnetically long solenoid coil, H electromagnetically an of using assumption the 1.2% carbon is more than three times lower times compared to the 1.2% three more is than carbon (768°C/1414°F vs. 732°C/1350°F). SAE steel carbon Curieof temperature plain 1008 1060 noticeably steel is from different content. of variation of carbon the because For Table example, from see one can as 3.3, the materials. Table steels. carbon for plain 3.3 shows of Curie some temperatures ferromagnetic the workpiece H surface chemical composition and microstructure. For example, composition the microstructure. and chemical cal carbon steel is shown in Figure 3.9a. Figure μ shown in is steel carbon maximum cal The tion diagram that illustrates the A the illustrates that diagram tion of a homogeneous carbon steel cylinder. steel carbon of ahomogeneous critical value ofcritical H piece’s 3.9b). core (Figure aresult, μ As the toward homogeneousat of work surface the the the workpiece exponentially off falls and as the field intensity in the air gap between the coil and the and theworkpiece. coil gapincreasing between air the With in fieldintensity the as µ Temperature, °C(°F) Magnetic Material Materials TemperatureCurie Magnetic of Some TABLE 3.3 . If H . If r su The magnetization curve describes the nonlinear relationship between magnetic flux magnetic between relationship nonlinear the describes curve magnetization The The temperature at which a ferromagnetic body loses its magnetic properties becom properties body its magnetic loses at temperature a ferromagnetic which The The maximum value of relative magnetic permeability value of relative permeability maximum The magnetic different be might Curie steels, temperature the Thus, carbon even plain among the Similar to the current distribution, the magnetic field intensity is at its maximum valueis fieldintensity at maximum its magnetic the distribution, current to the Similar A rf 2 corresponds to the surface magnetic field intensity H fieldintensity magnetic surface to the corresponds critical temperature. Figure 3.8b temperature. Figure shows critical Fe–Fe of the aportion → ∞, μ then and magnetic field intensity H fieldintensity magnetic B and B µ r (a) . When H> . When r

surf 1. In conventional IH of metals, the magnetic field intensity → 1.the field intensity at conventional magnetic In the of IH metals, B is typically much H that greater typically is 768 (1414) ) and μ ) and H 1008 cr H r versus field intensity ( intensity field versus cr B cr that corresponds to the maximum permeability is called a called is permeability maximum to the corresponds that , the magnetic permeability will decrease with increasing increasing with decrease will permeability magnetic , the 2 critical temperature being a function of carbon content of carbon afunction being temperature critical µ 732 (1350) r 1060 . The nonlinear variation of variation μ nonlinear . The r varies within the magnetic body. magnetic the At within surface, varies the H µ H μ of Hand (b) distribution ) and r (b Permalloy 440 (824) ) H Su cr r occurs at the “knee” of the curve. curve. of at the “knee” the occurs . rfa surf µ ce µ r ma µ r ma . In quick estimations under under estimations quick . In r ma x x is greatly affected by the by the greatly affected is x of high-carbon steel with with steel of high-carbon of low-carbon with steel 1120 (2048) Cor µ 3 surf Cobalt C phase transforma phase C r r =B e can be considered be can /( H r along the radius radius the along μ 0 ) for atypi 358 (676) Nickel and and 59 - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 intensity (A/in.) ture and magnetic field intensity is shown in Figure in 3.10a.Figure is shown fieldintensity magnetic and ture in induction billet heating. billet induction in workpiece 3.11 induction coil. Figure located an inside effect shows appearance of skin an ferent degree) whenever AC is there in a flow.be present will effect Therefore, the skin (though always which effect, occurs to skin adif the called is conductorthe section cross conductor within toward distribution its center. phenomenon current of nonuniform This of surface the the from decrease will physical density current properties;tromagnetic the always conductor of located surface be the will on the density homogeneous with elec value current of the maximum The conductor, not is uniform. distribution current the However, conductor’sthe uniform. is section cross AC when an same the flows through ductor alone (e.g., stands that within or bar cable), bus distribution current electrical the flows a con basicsthrough of one the mayAs electricity, current from know when adirect Effect Skin 3.1.2 3.10b). (Figure to decline sharply start Curie point would permeability magnetic distance from the surface, the μ from distance 60 company.) Group Inductotherm Inc., an of Inductoheat (Courtesy heating. billet induction in effect skin of the Appearance FIGURE 3.11 tively “weak” field, magnetic μ heat-treating of and majority IH applications, the In However, case. ture. it the is arela in netic permeability in relatively “weak” and in “strong” magnetic fields. “strong” magnetic in and “weak” relatively in permeability netic [1].magtemperature and of (b) Comparison intensity of field afunction as permeability (a) magnetic Relative FIGURE 3.10 begins to fall off (Figure 3.9b). (Figure off to fall begins complex The of μ nature Mag From Figures 3.8From 3.10a, Figures and conclude one might μ that 300 250 200 150 100 netic field 100 50 180 250 640 1300 750 600 r increases, and after reaching its maximum value at its H= maximum reaching after and increases, 450 r might first increase with temperature and only near the and near only with temperature increase first might Te 300 mp 150 erature (° 10 50 100 150 200 250 300 C)

r Relative Magnetic Permeability as a complex function of acomplex tempera as function r always decreases with tempera always with decreases Handbook of Induction Heating ofInduction Handbook “W “Stron ee k” fie µ g” r

Te = 1 field ld mp eratur e cr , it ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 the radius of a nonmagnetic metallic solid cylinder at two different frequencies F frequencies at solid cylinder different two metallic ofthe radius anonmagnetic example, an As along 3.13a Figure distribution density of current acomparison illustrates flow solid cylinder, no the current is inducedat in there centerits (Figure current 3.12). layersurface (“skin”) eddy of the nature workpiece. of the circumferential of the Because Background Theoretical Instantaneous distribution of AC currents in “coil-to-workpiece” in system. of AC currents distribution Instantaneous FIGURE 3.12 and current and power density profiles owing to skin effect (b). effect to skin owing profiles power density and current and (a) frequencies different at two cylinder steel of astainless radius the along distribution density Eddy current FIGURE 3.13 degree of skin effect depends on the frequency, on the depends ( properties effect material of skin degree face layer (or reference the conductor of penetration) the δ called depth is current that equation by the properties) roughly be calculated homogeneous can ing electromagnetic 3.13b).(Figure (assum workpiece along the density thickness current of the distribution The applied is or radius whenfrequency workpiece of the the relatively is compared large to δ when high effect tor, workpiece. of apronounced be the skin geometry the will and There This effect is one of the major factors that cause the concentration of eddy current in the concentration one major the of is factors the the cause of that in eddy effect current This Because of the skin effect, approximately effect, skin of 86% the Because sur the concentrated be power of in the will (a)

Su CurrentDensity rfa ce F 2 F

> F 1 F 2 1 C ylinder load Cor +I Axis ofsymmetr −I e core atanyf heat sourcesinth ere arenota R Induction coil re quenc Su y ny rfa +R e y II ce = 0 e (b − ) y

Relative value / 0.2 0.4 0.6 0.8 δ 0 1 , 01 (3.5) Coil Coil Su Loa rfa (in termsofcurrentp ce d –1 Distance fromsurface Cu rr 0 234 en R t ρ and μ and +1 enetration depth,δ J/ r ) of the conduc) of the surfac Cu density e density Po =0.368 rr wer 1 Cor en and F and t ) e γ/δ . The The . 61 2 . - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 of plain carbon steel SAE steel carbon δof 1040 plain the shows at (21°C temperature room or 70°F). substantially. Table Table 3.4 metals. shows nonmagnetic δof 3.5 used some the commonly cycle,value. increase heating the δcan Therefore, during metals, even for nonmagnetic δ layer of thickness clude 86% asurface and 63% that concentrated be current of power of the within the will However, 14% equal power the will density of its value. surface con we From can this, ( surface the from figure, the at onepenetration from depth towardsurface core. see the one can As powerworkpiece and the density from density of current by distribution showing ance

decrease to “1/expdecrease to “1/exp”exponentially will its value distance at power surface. The the at density this conductor toward decreases electrical of surface the itsthe core current at the which from Equation distance 3.5 the in depth is penetration the speaking, Mathematically relative permeability. and magnetic of root frequency square inverselyand the with Ω in ρis resistivity where electrical ( material where 550°C or 1022°F), μ of (at temperature rise temperature. approximately With ρwith afurther in increase of the 3.14a) slightly (Figure increase will steel carbon the into penetration current because the equations. those to define used (Figure 3.14a). (Figure temperature, significantly. above heating the After aresult, As δincreases Curie nonmagnetic. becomes where 62 value of relative magnetic permeability at the surface of the workpiece at ofvalue surface the the of relative permeability magnetic tion oftion μ distribu nonconstant of the because meaning 3.7)and not does have determined afully 3.6body, (Equations form its classical definition of δin to mention the that it necessary is at workpiece the (A/m surface the Curie temperature or Curie Curie temperature point,the μ δ is penetration depth (m). depth penetration is While discussing the three-dimensional (3-D) behavior of μ three-dimensional the discussing While its initial times to to four six increase cycle, can heating ρof the most the metals During As one can see, the value see, the one can ofAs δ is a function of temperature. At the beginning of the heating cycle, heating of the of stated temperature.As earlier, At beginning the afunction δis Penetration depth is described in meters as meters in Penetration described is depth I = frequency (Hz [cycles/s]), (Hz F =frequency conductive electrically of the resistivity ρ=electrical is the current density at distance y at density distance current the is r within the workpiece the practice, even the within at engineering ambient temperature. In Ω *m), μ and y with temperature temperature ρwith in increase of slightly the because continue to rise δ will =δ ), the current density will equal approximately equal 37% will density ), of its value. surface current the r 2 starts to markedly decrease. Near a critical temperature A temperature acritical Near to decrease. markedly starts . ” its value at 3.13b surface. Figure the appear effect askin illustrates r = relative magnetic permeability, or in inches as =relative permeability, magnetic inches or in 2 ), ), y is the distance from the surface toward surface core the (m), the from distance the is and varies with the square root of electrical resistivity resistivity of root electrical square the with varies *in. δ δ r drastically drops to unity because the carbon steel steel carbon the because to unity drops drastically = = 3160 from the surface (A/m surface the from 503 µ µ ρ r ρ r F F , , (3.6) (3.7) Handbook of Induction Heating ofInduction Handbook r 2 within a ferromagnetic aferromagnetic within ), ), I 0 is the current density density current the is µ r su rf is typically typically is 2 , known as as , known - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 Theoretical Background Theoretical

TABLE 3.4 Penetration Depth of Nonmagnetic Metals (mm)

T ρ Frequency (kHz) Metal °C °F μΩ·m μΩ·in. 0.06 0.50 1 2.5 4 8 10 30 70 200 500 Aluminum 20 68 0.027 1.06 10.7 3.70 2.61 1.65 1.30 0.92 0.83 0.48 0.31 0.18 0.12 250 482 0.053 2.09 15.0 5.18 3.66 2.32 1.83 1.29 1.16 0.67 0.44 0.26 0.16 500 932 0.087 3.43 19.2 6.64 4.69 2.97 2.35 1.66 1.48 0.86 0.56 0.33 0.21 Copper 20 68 0.018 0.71 8.81 3.05 2.16 1.36 1.08 0.76 0.68 0.39 0.26 0.15 0.10 500 932 0.050 1.97 14.5 5.03 3.56 2.25 1.78 1.26 1.12 0.65 0.43 0.25 0.16 900 1652 0.085 3.35 19.3 6.67 4.72 2.98 2.36 1.67 1.49 0.86 0.56 0.33 0.21 Brass 20 68 0.065 2.56 16.6 5.74 4.06 2.56 2.03 1.43 1.28 0.74 0.48 0.29 0.18 400 752 0.114 4.49 21.9 7.60 5.37 3.40 2.69 1.90 1.70 0.98 0.64 0.38 0.24 900 1632 0.203 7.99 29.3 10.1 7.17 4.53 3.58 2.53 2.27 1.31 0.86 0.51 0.32 Stainless steel 20 68 0.690 27.2 53.9 18.7 13.2 8.36 6.61 4.67 4.18 2.41 1.58 0.93 0.59 800 1472 1.150 45.3 69.6 24.1 17.1 10.8 8.53 6.03 5.39 3.11 2.04 1.21 0.76 1200 2192 1.240 48.8 72.3 25.1 17.7 11.2 8.86 6.26 5.60 3.23 2.12 1.25 0.79 Silver 20 68 0.017 0.67 8.34 2.89 2.04 1.29 1.02 0.72 0.65 0.37 0.24 0.14 0.09 300 572 0.038 1.50 12.7 4.39 3.10 1.96 1.55 1.10 0.98 0.57 0.37 0.22 0.14 800 1472 0.070 2.76 17.2 5.95 4.21 2.66 2.10 1.49 1.33 0.77 0.50 0.30 0.19 Tungsten 20 68 0.050 1.97 14.5 5.03 3.56 2.25 1.78 1.26 1.12 0.65 0.43 0.83 0.53 1500 2732 0.550 21.7 48.2 16.7 11.8 7.46 5.90 4.17 3.73 2.15 1.41 0.83 0.53 2800 5072 1.040 40.9 66.2 22.9 16.2 10.3 8.11 5.74 5.13 2.96 1.94 1.15 0.73 Titanium 20 68 0.500 19.7 45.9 15.9 11.3 7.11 5.62 3.98 3.56 2.05 1.34 0.80 0.50 600 1112 1.400 55.1 76.8 26.6 18.8 11.9 9.41 6.65 5.95 3.44 2.25 1.33 0.84 1200 2192 1.800 70.9 87.1 30.2 21.3 13.5 10.7 7.54 6.75 3.90 2.55 1.51 0.95 63 Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 These thermal gradients result in nonuniform distribution of ρ distribution nonuniform in result gradients thermal These heated the workpiece. within gradients always are there thermal and not is uniform tion distribu density following sections), the current in the discussed phenomenanetic are that electromag some other of and applications majority IH great for the effect (owing skin to the ρ solid body constant for only with ahomogeneous correct nonmagnetic is assumption workpiece. the into surface the from However, decreasing exponentially being as this workpiece distribution) along the source thickness/radiusintroducedand simplified are applications.eration heating for through phenomenon consid into this to take important cycle. heating It especially is atypical ing 3.14b Figure effect. dur of skin increase can δof shows some the metals how times many Penetration Depth of Carbon Steel SAE Steel 1040 of Carbon at Depth Temperature Room Penetration (21°C or 70°F) TABLE 3.5 64 for different metals during IH (b). IH during metals different for ( depth penetration of current Typical variation FIGURE 3.14 A/mm sity (Figure 3.9b). (Figure sity distribution density current of exponential Therefore, assumption an fieldinten magnetic of the workpiece distribution ferromagnetic to anonuniform owing 10 40 80 160 Magnetic Field Field Magnetic 200 280 (a)( Realistically speaking, this assumption can be made for only some unique cases because because made be cases for some can only unique assumption this speaking, Realistically power most and publicationsIn devoted (heat of densities current to IH, distributions degree the workpiece steel of IH acarbon changes of variation δduring The drastically In addition, as shown in the previous section, μ previous addition, section, the In shown in as Current penetration depth (δ) Intensity A/in. 1000 2000 4050 5100 7100 “ 250 Cold” stage Tr ansient stage 11.20 mm 2.50 4.70 6.30 8.76 9.63 Te mp 60 erature 0.100 0.185 0.249 0.345 0.379 0.442 in. mm 0.88 1.63 2.20 3.03 3.33 3.89 “Hot” stage Cu 500 rie p 0.034 0.064 0.086 0.131 0.153 0.119 in. oin δ t Frequency (Hz) Frequency ) during IH of a carbon steel workpiece (a) and variation of δ (a) workpiece steel variation and of acarbon IH ) during mm 0.36 0.67 1.24 1.36 1.59 0.9 3000 Penetration Depth Penetration b) 0.014 0.026 0.035 0.049 0.054 0.062 in.

r Increase of current penetration depth is nonuniform along the thickness of the of the thickness along the nonuniform is (times) 10 12 14 16 0 2 4 6 8 St mm 0.36 0.49 0.68 0.75 0.87 0.2 . St 10,000 ee lB Handbook of Induction Heating ofInduction Handbook 0.008 0.014 0.019 0.027 0.029 0.034 in. and μ and ra ss mm 0.21 0.28 0.39 0.43 0.50 0.11 r Al within the workpiece. the within 30,000 0.004 0.008 0.015 0.017 0.020 0.011 Cu in. W1 mm 0.06 0.12 0.16 0.21 0.24 0.27 100,000 040 0.002 0.005 0.006 0.008 0.009 0.011 in. . - - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 cally cally electromagnetically electromagnetically fieldandan couldbe considered electromagnetic to the it semitransparent then becomes solid of body, the or diameter field compared thickness greater to orientation.If δ is the magnetic and frequency chosen on the depending bodies “act”piece or can thin thick as sections. corresponding in text this later etc.) applications in heating, bar different (surface discussed hardening, is [1,2,17–21,44,52–54]. effect skin of the nature true the ing phenomenon in impact of this An physics the regarding of power that prediction wave-shaped density phenomenon reveal flow. current the disturbing without its solid heating body the during within power the distribution measure density not possible to reveal phenomenon Obviously, that more detail. in it was not possible to to accurately IH the ofsoftware modelprocesses, sufficientlysophisticated it ability was Unfortunately, unavail the and at of time computer that because limitations modeling form. (heat exponential sources) assumed would commonly compared to the different be power ofintuitively the should felt when situations there be density distribution that [7,8]. texts respective power their and in of densities current distribution authors Both decline. its final it starts amaximum, reaching after and again, to increase surface, power the the from starts density distance a certain toward power core. surface. the the Then, the decreases However, density once it reaches power located is at density maximum The distribution. exponential assumed commonly the from radius/thicknessdifferent the wave-like significantly aunique is form, which has only. materials nonmagnetic offor heating estimations of rough engineering (Equation forpurpose the 3.5) used be can Background Theoretical will be considered an electromagnetically thick electromagnetically considered be an will of 12.7 billet (0.5 steel mm contrast, in.) In astainless Hz. lower 500 diameter frequencies than thin electromagnetically an as IH (1 of act during 25 mm billet in.) will diameter tromagnetically a thin on platedepending orientation as inductor, compared to the act electromagnetically it can thick electromagnetically an called be it can otherwise, workpiece the considered be then athin cancellation, eddy can in current results cally fieldnetic maythe have on workpiecean electromagneti considering as marked effect a magnetically thin magnetically thick electromagnetically an cycle, could considered heating be workpiece as the the during 15 initially that times than thick electromagnetically considered an be it as then can thin electromagnetically in appearing the amounts heatof generation small be only field,will there electromagnetic external the inductors traveling wave and exceptions). inductors to are transparent practically Being magnetically magnetically When discussing the skin effect, it effect, appropriate is electromagneti skin the terms to the introduce discussing When new generationThe of a quite allows accurate one computer to make software modeling P. and wave-shaped independentlyM. suggested Lozinskii this originally Simpson along power the some distribution applications, density In hardening, surface as such Geometry alone cannot determine whether aworkpiece determine should elec considered be an alone cannot Geometry Besides the frequency,Besides the orientation workpiece electromag of the to the the respect with δ the conductive times solid of body electrically six the is or diameter thickness the If or thick or thick thin or thick or and electromagnetically thin electromagnetically and body. thin thin body that is accompanied by a dramatic reduction in heating efficiency. heating reduction accompanied by in body is adramatic that bodies absorbing only a negligible amount of energy (transverse anegligible only absorbing of amount flux energy bodies body. orientation solid If of workpiece the inductor compared to the or thick or thin body. There is a distinct current cancellation within the electro the within cancellation body. current adistinct is There body. For example, steel asolenoid when using coil, astainless body can become at the end of the heating cycle an electro cycle become an at heating end body of the the can bodies. bodies [1]. bodies awork speaking, Electromagnetically body if a frequency of 70 kHz is applied. of is 70 afrequency kHz body if body. more increase δcan Since body. 3.15 Figure shows that body using using body body; body; 65 ------, Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 opposite directions and tend to cancel each other. This phenomenon is used in coaxial cables. coaxial in phenomenon other. each used tend to is cancel and This opposite directions fields generated magnetic noticeablyconductoreach by external case, the have weaker. this In be fieldwill magnetic external the areas, internal concentratedthe is in other. current the Since complementingeach direction fieldsame the are produced having magnetic bar each bus by of lines 3.17a). (Figure imaginary opposite the area, in directions this in because occurs This concentrated be on oppositewill sides conductors of the 3.18b). (Figure areas). (internal other currents However, these then have direction, same the currents the if 3.17b) (Figure each facing currents areas both the concentrated be then in will directions, the conductorsin have flowing opposite currents the If redistribute. conductorsboth will body clearly 3.16b. is visible on Figure arectangular in effect ofappearance skin conductor (e.g., (b) a rectangular in distributions density bar) bus alone. stands The that placedtor is given nearby below. is 3.16 Figure field shows magnetic the (a)current and 3.1.3 66 nearby fields, affecting the current flow and current distributions. the power density nearby fields,affecting proximity.tors in conductors fields,interactwith These have magnetic own which their conduc other are Most case. not there most applications, often, is the practical In area. this conductors surrounding the in current-carrying no other are there that alone and stands conductors in or effect cables, a conductor skin that it the was assumed discussing When thin of electromagnetically concept of the Illustration FIGURE 3.15 A strong magnetic field forms in the area between the bus bars when the currents flow currents the thewhen bars bus between area the in field magnetic forms A strong in conductor one,currents placed first the is the near current-carrying another When An analysis of the effect on current distribution in aconductor in conduc when distribution another on current effect of the analysis An Electromagnetic Proximity Effect Electromagnetic Ele (there iscurrentcancellation) ctr omag M inducto Slab/plat ultitur netically Y r n e thin Eddy curren Y body X t and thick and Z X (there isnocurrentcancellation) Elec bodies. Elec Elec tromag tromag tromag thic thin k body Z netically body Handbook of Induction Heating ofInduction Handbook netically netically thic k body X - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 rents are flowing in the opposite directions. opposite the in flowing are rents cur the when effect the proximity to owing bars bus in (b) distributions density field current (a)and Magnetic FIGURE 3.17 Magnetic field (a) and current density (b) distributions in a stand-alone rectangular conductor. rectangular stand-alone in a (b) distributions density field current (a)and Magnetic FIGURE 3.16 currents are flowing in the same direction. same the in flowing are currents the when effect the proximity to owing bars bus in (b) distributions density field current (a)and Magnetic FIGURE 3.18 Theoretical Background Theoretical shown in Figure 3.19. Figure shown in is systems of nonsymmetrical case the in effect Proximity decrease. will effect proximity complementing other. each direction field of two magnetic producedhave the conductors lines same theby will imaginary the bars; bus however, the because be quite strong between fieldwill magnetic external the 3.18a). occur a relatively cancellation, field(Figure will weak of magnetic this Because other each canceling bars bus between area the in have opposite will directions lines If the distance between conductors increases, then the strength of the electromagnetic electromagnetic of the strength the conductors then between increases, distance the If field magnetic the then have direction, same the currents the opposite if The true: is (a) (a) (a) (b ) (b (b ) ) 67 - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 of an inappropriate appearance of the proximity effect. Figure 3.21a,b Figure effect. inappropriateof an proximity of appearance the shows of result the because occurred melting. 3.21 that Figure heating examples of shows nonuniform three cold hot and spots, lead overheating, to can localized which even effect, and proximity the undesirable the of appearance could in result mechanism recipe, workpiece and handling Inappropriate or beneficial. inductor combination of design, process harmful be might noticeably coupling.good be heat wider shallower 3.20b). Also, will the and pattern (Figure of thecase in as significant not as be coupling), will rise electromagnetic temperature the gap (poor air larger the relatively be with deep. area and heat the narrow will In pattern there. The heating coil-to-workpiece intense (good coupling), an gap small in is air resulting flow. eddy the current in by caused adifference tion). is patterns of appearance these The heated statically (without developed is be cylinder the if will rota noticeably patterns different inductor. Two asingle-turn inside located is asymmetrically cylinder steel Acarbon terns. factor that causes a current redistribution in an IH system as shown in Figure 3.12. Figure shown system in IH as an in redistribution acurrent factor causes that second the is other. each facing This areas workpiece concentrate the the induced in in will eddy and inductor. currents current coil the effect, Therefore, proximity of aresult the as workpiece of the the current compared source of haveto that the opposite an direction piece inductor. located the near of Faraday’s Because law, induced within eddy currents 3.12), (Figure current source conductive the carries work electrically the is other the and ofsist at least conductors two [1,6,7,44,54,55]. conductors inductor of that an these One is 68 Proximity effect in single-turn coil with nonsymmetrical positioning of the workpiece. of the positioning nonsymmetrical with coil single-turn in effect Proximity FIGURE 3.20 systems. nonsymmetrical in effect Proximity FIGURE 3.19 Depending on application specifics, the presence of an electromagnetic proximity effect effect proximity on applicationan electromagnetic the Depending of presence specifics, where the area 3.20a, the Figure have in shown eddy in As density the ahigher currents Figure 3.20 shows how the electromagnetic proximity effect produces different heating pat produces heating effect 3.20 different Figure proximity shows how electromagnetic the The phenomenon of proximity effect is directly related to IH. con Induction systems directly is phenomenonThe effect of proximity Coil ax symmetr curr Eddy en is of t Op y po sit Single- ec coil urrent tu rn sc Po Simila wer density r Handbook of Induction Heating ofInduction Handbook urrent s - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 the current will occur on its opposite occur side [1,6,7,44,51]. will current the sides on the conductor; of the distributed be however, will current the of portion asmall the flow the conductor surface of thatthe near rent faces will workpiece. of Theremainder 3.22a). (Figure there 3.22b, Figure shown in As appreciable an inductor’s of the portion cur flux density magnetic workpiece, highest inductor the the the and in between resulting workpiece) 3.22). (Figure placed itsthe proximity is gap in in fieldsqueezed is Magnetic conductivebody an electrically when redistribute (e.g.,will density fieldcurrent andnetic conductor 3.16). (Figure astand-alone current-carrying in distributions density mag The current field and wemagnetic effect, reviewedthe first proximity the weWhen discussed 3.1.4 power efficiency. overall improving energy thus losses, and transmission reducing and drop voltage minimizing impedance, its decrease significantly proper of network design abus will power in but IH supply not also in only important network bus and The design design. patterns. heating required of obtaining ameans as effect inductors ing).and rely(Figure single-shot 3.2) on a Profiled proximity coils primarily reliev (e.g., stress and areas selective heating tempering, localized hardening, selective heating. workpiecelar scan during 3.21c coil.induction Figure at spots of “hot” edge adeformed tubu areas shows localized an through its processing during handling to inadequate tube owing effect a proximity severe overheating by caused oflocalized copper tubing (a) wall (b) thin aluminum and Background Theoretical is placed in proximity to a rectangular conductor. arectangular to proximity in placed is body conductive (e.g., workpiece)electrically an when (b) redistribution density field current (a)and Magnetic FIGURE 3.22 spots at edge areas when heating a deformed tubular workpiece. tubular adeformed heating when at areas edge spots “hot” (c) and coil localized induction an through processing its during handling tube improper to owing effect by aproximity caused (b) tubing (a) copper aluminum wall and of thin overheating severe of localized Results FIGURE 3.21 An understanding of the physics of the electromagnetic proximity and skin effects is is effects skin and proximity physics of the electromagnetic of understanding the An in amajorfactor beneficial is effect proximity electromagnetic an hand, other the On Electromagnetic Slot Effect Electromagnetic (a) (b (a) ) (b ) (c) 69 - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 components (e.g., devices). or electronic fixtures, cabinets, conductive electrically of certain shielding to have electromagnetic when it an required is motor as such generators, cases machines in AC and industrial machines, other DC and diameters. internal heating split-return inductors, single-shot, and inductors well as as for hairpin, channel, role in conductor of the concentrator. properties geometry, the and electromagnetic and conductor the frequency, in on the depends fieldintensity, distribution magnetic current “open the concentrated be in but still slot. most of surface” of the actual area it The will conductor, the in slightly be redistributed 3.24). piece (Figure will current case, the this In electromagnetic coupling that improves the electrical efficiency IH.of efficiency coupling improves that electrical the electromagnetic improvedbe inductor-to-workpiece will phenomenon, there this to field. Thanks magnetic reduce helping external and to dramatically of areas selective heating the widely in used slot is effect electromagnetic slot The effect. electromagnetic an phenomenon called is “open to the tor current slot. surface”of concentrator—to of the area the open This the workpiece 3.23b). (Figure words, concentrator other induc the In magnetic squeezes the 3.23a. Figure concentratormagnetic provides alowin field path reluctance as forshown amagnetic conductor, this C-shaped 3.23. around Figure shown laminations) in as a result, As the 70 Slot effect takes place without the presence of the workpiece in the stand-alone conductor. stand-alone the in workpiece of the presence the without place takes Slot effect FIGURE 3.24 concentrator. “open of the (b) the to surface” current inductor the fieldsqueezes and magnetic (a) a for path a reluctance low provides concentrator flux Magnetic FIGURE 3.23 The slot effect is widely used not only in connection with IH but also in the design of design the in but IH with also connection widely not in slot is only The used effect play important effect aparticularly proximity the slot and effect electromagnetic The It is necessary to mention here that the slot effect will also take place take without work the also slot will to mention the here that effect It necessary is conductor’s of the the all concentrated be facing surface Practically on the will current study, our flux concentratorContinuing magnetic external position let an (e.g., us (a) (b ) Handbook of Induction Heating ofInduction Handbook - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 Ring effect in a round conductor. around in effect Ring FIGURE 3.26 conductor. arectangular in effect Ring FIGURE 3.25 nation with the skin and proximity effects, it will lead to a concentration it current effects, coil of will the proximity and skin the nation with plays effect solenoidlocated the apositive inside combi coil, induction this role in because efficiency.process For example,in conventional cylinder solidthe IH workpiecewhen is 3.12. Figure system induction shown in the in tion distribu responsible current for is that the effect electromagnetic Therefore, third it the is coils. multiturn inductors in but also place single-turn not takes in only effect ring The coil. induction of surface the ductors inside on the to aconcentration leading of current tions (thus other). to each attracted tions being opposite opposite oriented of in the direc surfaces are sides ring’s of the circumference inside flowing on currents because effect proximity to the somewhat also similar is effect lowest the path and [1,2,6,7,54]. impedance shortest distance the ring see, this one can As be insidewill layer there surface where ring the of thin the flowwithin will current of the result,As a most disbursed. be will lines flux magnetic the there. Outsidedensity ring, the flux magnetic increasing ring, the insidebe concentrated will lines uted. flux Magnetic redistrib be will its current bent then to shape is bar it aring, into current-carrying that Figure 3.25. If shown in are bar, bus conductor, distribution its current and arectangular Up conductors. straight to now, in such One distribution density current we have discussed 3.1.5 Background Theoretical The presence of the ring effect can have can apositive effect or negative ring of presence the and The heating impact on the Figure 3.26 shows the appearance of the electromagnetic ring effect in cylinder con cylinder in effect ring 3.26Figure shows electromagnetic of appearance the the Electromagnetic Ring Effect Ring Electromagnetic 71 - - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 or chemical/powder [1,44,57]. guns are achievable higherthan by rockets to velocities materials fire can or launchers guns tric forces, to toa capability developFor elec example, electromagnetic large incredibly thanks values. tremendous reach some forces applications, can In springs. ings, and electromagnetic pumps, levitators, bear electromagnetic seals, electrical motors, magneto-hydro-dynamic including technologies modern forces many role play in important an Electromagnetic Force Electromagnetic 3.1.6 circuits. cooling the ing phenomenonavoid account into this when design to take overheating, local it necessary is of appearance spots. To “hot” and buses of the undesirable areas overheating of localized to bent, leading are where potentially buses concentratedareas is in current the effect, this 3.27b). (Figure effect ring efficiency, coil the measurably(butand increase not completelydominate the eliminate) to improve coupling,slot to “assist” effect effect equivalent proximity electromagnetic the allows applications, a heating flux concentratorThis the insidecoil. amagnetic is located consumption. energy and kilowatt copper coil losses increasing drastically and efficiency electrical coil reducing workpiece, the from current coil de-coupling of dramatically the in results Thus, this to flowinsidediameter. current on coil of forces majority the and its effect proximity overpowers usually coil the the in appearing effect ring electromagnetic the diameters, For moderate effects. ring and and, especially, proximity the for relatively inside small phenomena: applications such electromagnetic of a result in counteracting two is tion efficiency. process desirabledistribu highly forTherefore,is current increasing the coil outside to an coil, of surface which current the coil the to here tends shift effect proximity coupling efficiency coil and,[56]. therefore, decreases However,the effect, ring despitethe equivalent coil-to-workpiece an worsens coil. of This the electromagnetic diameter inside 3.27a). concentration on the leads tocurrent acoil effect ring case, the (Figure this In I.D. (so-called ters heating), where hollow inductor located the the is inside workpiece electromagnetic coupling, which leads to a coil efficiency increase. coupling, leads toefficiency which acoil electromagnetic improved be coil-to-workpiece aresult, will coil. of As the there diameter inside on the (b). concentrator flux a“U”-shaped magnetic (a) with coil acoil abare versus using when tion distribu field magnetic in difference the Compare diameters. inside heating when effect ring of the Appearance FIGURE 3.27 72 The ring effect should be taken into consideration into should power taken be in effect supply ring The of Because design. of majority I.D. agreat in effect, order ring the In to dominate “help” effect proximity the The ring effect plays diame or effect inside surfaces anegative ring The of IH internal the role in (a) (b ) Handbook of Induction Heating ofInduction Handbook ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 IH applications. IH conductors. greatly on orientation current-carrying of the amongthat factors, force field.other is that forces magnetic here depends from The lesson any field,not experience then magnetic will it conductor external to an parallel is carrying conductor. words, other current- In the current-carrying by if the experienced be force will φ=0and, to zero, therefore, field sin equal no Bis then magnetic Iand current oftion the Left-hand (FBI) rule of magnetic force. of magnetic rule (FBI) Left-hand FIGURE 3.28 force. of the show direction the will palm),thehead lines into thumb the field magnetic then field external (imaginary the flux of the magnetic of fingerdirection flowthe pointer and follows ofthe tion current (FBI rule). rule followshand direc thethe middle left finger of the rule, According if to the left-hand on the based determined field be Bcan magnetic conductor external placed an in element by the force of d the experienced shows direction the that B flux density magnetic where aforceence d element d Biot-Savart and Ampère current-carrying If a [1,44,58–65], quantified. be force can this study by an experimental to flux density. magnetic and Thanks to current proportional topic. to by the providing abrief introduction remedy this paragraph intended to forces. at is todures evaluate This least partially electromagnetic 3.2), (Figure geometry coil to itdevelop difficult makes which butgeneral simple proce of workpieces variety specific endless heated by require induction Many seemingly of the [1,44,57].quality may which negativelyture, overall system reliability, affect repeatability, heating and workpiece, or fix concentrator, or fixture, flux turns inductionand the bend coil even system Without design. heated proper the consideration, reallocate forces can those and that have producelife forces on coil significant a currents effect pronounced High hardening. gear and for some not shaft unusual are higher and kA of 3–5 currents Background Theoretical Let’s forces in of appearance of consider magnetic the some most of cases common the between the direc the φbetween Equation angle from 3.8 to the remember It if that important is Newton (1 in force measured SI the N=0.102 is units, In =0.225 kgf lbf). 3.28 Figure A current-carrying conductorthatis a fieldforce experiences placed a magnetic in A current-carrying publications. IH in Unfortunately, discussed not often forces so are electromagnetic values. For substantial reach example, also applications, can IH many In currents coil F , I and Iand current of the direction the between angle the vectors φis and Bare , and l , carrying a current I acurrent , carrying F according to Equation 3.8: Equation to according carr Element ofcurrent . ying conducto , is placed in an external magnetic field B magnetic external placed an , is in dd FI r =× - Bl = IB I d dl l F si n, φ ϕ B

of the current-carrying current-carrying l of the , it will experi , it will (3.8) 73 - - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3

74 Magnetic interaction between two thin wires. thin two between interaction Magnetic FIGURE 3.30 conductors. current-carrying in force Magnetic FIGURE 3.29

where 2. 1.

netic force experienced by the second wire, according to according Equation wire, second by 3.8, the force be netic experienced will attempting to conductors, separate the attempting F 3.29a), (Figure opposite the forces direction oriented in experience are will which are other,each located conductoreach near then in oppositedirections flowing conductors (such or cables) buses as currents current-carrying two If having They will experience an attractive force, an experience will They 3.29b), conductors (Figure tion the to bring together. try forces will resultant the direc same the oriented in currents contrast, conductors two In if carrying are to Equation 3.9: produces field amagnetic according wires current-carrying parallel ofeach the [1,44,58–65],have orientation. same According to the basic electromagnetics of (0.8 20 of separated mm by 200 Aand adistance current in.). currents Both a carrying each wires, thin two between forces occurring attractive magnetic of follows deform What bars. they bus calculations that simplified large so are is the radial distance between the wires (Figure 3.30). (Figure wires the Therefore, mag between the distance radial the R is (a) B -field (1) (2) I 1 FI B = = F 2 1 2 µ µ 2 π 12 0 π R 01 I =− I R F , 12 (3.9) l R =F F (b 21 ) (1) . 21 F . In some cases, the forces are the some cases, . In 2 F I 21 2 Fo Handbook of Induction Heating ofInduction Handbook rc e F 12 (2) - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3

Background Theoretical Magnetic forces in an empty solenoid coil. solenoid empty an in forces Magnetic FIGURE 3.31

and the force per unit length will be will length force unit the per and is the number of turns in the long the solenoid l in of length of number turns the N is In this case, the force per unit length will be will length force case, unit the per this In 3. tangential component of vectortangential H(magnetic fieldintensity). In the case of the infinitely long multiturn solenoid, long H multiturn infinitely of the case the In inside the long homogeneous as the and inside solenoid expressed be can sure f sure pres magnetic tensile radial The opposite the direction. oriented in is turn each the opposite on side of circumferentially flowing current the because direction, radial the forces in solenoid of the tensile experience turns the time At same the that the longitudinal magnetic pressure (density N/m force pressure in magnetic of magnetic the longitudinal the that longsolenoid and infinitely a field,homogeneous an magnetic be can shown it compressive Assuming longitudinal stresses. experience will turns the direction, same solenoid the oriented in multiturn are of the turns to neighboring respect in 3.31). forces (Figure turn electromagnetic each in ing flowing currents the Since voltage it, flow within produc applied solenoid acurrent in to amultiturn results applied solenoid be to also amultiturn inductor. phenomena can These Alternative R can be described as: described be can F l F = 1 (( 41 π fF ll ×× == 0 fH − Rt /Are 72 Wb == 20 π µµ (. a/ /A 0 F l HN 02 = 22 t µµ // = I F F m 0 22 2 R R HB mA ) µ 2 tt 22 Il π )( 01 B / I R 200 22 t (3.11) . = () )) 0 / () . = (3.12) 04 0 . . t is the root mean square square mean root the is

N/ is the coil current. current. coil the Iis , and F m. 1 2 ) (3.10) f - - l

75 Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 material properties, heating mode properties, heating (constantmaterial power, voltage), current, parameters. other and on frequency, depends also cycle,ing force power the distribution density, temperature/ heat the During not system, of is constant. the and geometry of the only notis afunction cycle heating the of forces during distribution orientation 3-D and the that to remember important the others. is It than greater radial, significantly or be hoop—may longitudinal, force components— three one specifics, process of and design on coil Depending 76 Magnetic forces in bar end heating of magnetic (b) and nonmagnetic (a) bars. nonmagnetic (b) and of magnetic heating end bar in forces Magnetic FIGURE 3.32 In most IH applications, the electromagnetic force has acomplex force most applications, has IH In distribution. 3-D electromagnetic the

4. Two 3.32. Figure shown examples in are typical 3.1.7). Section in discussed are edge forces end and (electromagnetic effects netic mag of the have distribution and magnitude effects apronounced impact on the andedge case),end the realistic the electromagnetic is length of(which finite are workpiece.long nonmagnetic However, workpiece and coil induction when the longsolenoid infinitely and infinitely considered an only has far so discussion The (e.g., resistivities low having heat electrical metals Cu, alloys). Mg, Al and inductors) butterflysplit-return, and relatively and using when to frequencies low (e.g., heating rely on proximity for inductors primarily that particularly pancake, consideration, into should taken be pressure magnetic reliable design, this and coil workpiecewhereas the compressive under is order In pressure. to provide arigid pressure, magnetic tensile experience turns coil site the current, coil of to that the oppo heated the workpiece adirection oriented in within are coil induction by the resistivity. induced It eddy electrical since stant that must currents emphasized be with a loadcon long nonmagnetic infinitely ofsolenoid an or consists empty is 4.1. 4.2. Another assumption used when deriving Equations 3.10 Equations when deriving used assumption Another 3.12 and the that is the two. the stronger of the typically is to force magnetization coil.of The attributed the toward bar middle the the attempts to pull which effect, magnetization the attempts to from remove inductor, the from bar resulting other the the and which effect, demagnetization the of from forces:bination two one resulting com a is force inductor 3.32b). (Figure placed amultiturn inside resulting The partially However,is bar when aferromagnetic situationquite different is the metals. resistivity of low electrical coil. Usually, the from bar the to eject stronger forces bars when result heating 3.32a) to provide, try force will for example, magnetic the end heating, bar the inductor (Figure placed amultiturn inside partially is bar a nonmagnetic If (a) Net forc eN S inductor olenoidal Nonmagn cy linder etic (b ) et forc e S inductor olenoidal Handbook of Induction Heating ofInduction Handbook Mag cy linder netic - - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 temperatures below abovetemperatures and Curie point. the apower/frequency using diameter) shaft of combination 180 kW/1 [57]. kHz a2-in.-diameter (50-mm- for coil induction (right hardening images) atwo-turn around design. to develop should taken be actions reliable which robust and coil/fixture to determine and to help accurately forces designer the evaluate required is electromagnetic modeling the For of applications majority IH cases. complex the having computer numerical geometries, etc.). separators, tromagnetic levitators, locks, pumps and induction plasma, electromagnetic elec furnaces, melting in (e.g., process of the part able technological play and important stirring an electromagnetic noise. However,intensive vibration industrial and desir be forces can those cases, other in Background Theoretical Park, Ohio, July, 2005.) 25–28, Treating Progress Heat Series, Induction Professor IH, in V. Rudnev, forces Electromagnetic (From respectively. 180 were (a) kW 1kHz, point and above it frequency (b). versus applied power and Coil Curie the below heated is shaft when forces of axial a magnitude Compare shaft. a steel heating when coil tion induc two-turn by a experienced forces major and distribution field magnetic of plots the Computer modeling FIGURE 3.33 Table 3.6 lists the magnetic forces produced when this carbon steel shaft is heatedTable is to shaft steel forces carbon produced magnetic when 3.6 the this lists StudyCase applied be some given specific/ formulas here can in only the that mind in Bear system, induction of the causing rigidity to the harmful be Excessive forces can magnetic Force Component Force N(lbf) (2-in.-diameter) Shaft, Steel a50-mm-diameter IH When Produced Forces TABLE 3.6 F Source: F F radial longitudinal hoop . An example given in Figure 3.33 example field Figure shows. An distribution given in magnetic the Treating Progress, ASM Int’l,Materials Park,Ohio,July, 25–28,2005. V. Rudnev, Electromagnetic forces ininductionheating,Professor InductionSeries,Heat (a) T T (b wo- wo- ) coil coil tu tu rn rn 209 lb 99.6 lb 99.6 lb 209 lb Shaf Ab Be t low Cu ove Cu Below Curie Point Curie Below rie temp rie 443 (99.6) temp 81 (18.2) 1.4 (0.3) eratur eratur symmetr Axis of e e

y Mag Mag field field neti neti c c Above Curie Point Curie Above 928 (209) , ASM Int’l, Materials Materials Int’l, , ASM 5.2 (1.2) 298 (67) simplified ­simplified 77 - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 viewed as an area consisting of three zones: a central part, a transverse edge effect area, area, edge effect part, zones: atransverse acentral of three consisting area viewed an as be the can slab fieldin electromagnetic the its thickness, much than width larger are and field (Figure magnetic 3.34).the slab’s uniform If initially placed an a slab is in length Refs. [1,2,9,44,76–82] in reported others. and been has early edge 1970s and effects end effects [75]. of electromagnetic analysis Further out by was D. carried end effects Lavers late the 1960s of in electromagnetic and analysis devoted study. been much has to their effort first to attempt The provide a systematic effects, of trapezoidal workpieces. these and importance great of the rectangular, Because cylindrical, in profiles temperature responsible for nonuniform primarily are by them caused fielddistortion the and effects These edge end and to effects. as referred are rents induced cur of distributions and corresponding fielddistortions edge Those and regions. by,the coil’s fieldaffected in factors, electromagnetic of among other the end adistortion profilesalongthe temperature The workpiece’s effect. skin by the width andare length 3.1.2, greatly affected Section is in difference temperature surface-to-core the discussed As 3.1.7 magnitude. to reduce their taken measures and systems IH forces when must designing addressed be orientationand of electromagnetic desirable, be forces positive can process. Magnitude those important rolethe playing an in excessivesive in acoustic vibration, noise. However, resulting and applications, other in exces and fatigue cracking to stress owing failure coil system, IH ofpremature causing an repeatability and rigidity the appreciable reach adverselyforces can magnitudes affecting properlyforces. magnetic ficient turns addressing support to coil should effort made be to provide systems. Thus, IH special eration when asuf designing should and properly be consid into life coil taken affects it since dramatically neglected force of 928 N(209 longitudinal lbf).maximum Obviously, be intensive such force cannot 78 January, 29–34, 1995.)January, 29–34, magic,” “black 1, Part is no longer andstrips bars IH of slabs, flux D. Longitudinal Loveless, (b). slab of V. the (From effect edge and Rudnev, end slab, for electromagnetic and system (a) of coordinate Sketch FIGURE 3.34 It is convenient to introduce these effects studying the IH of a rectangular slab. Suppose of IH arectangular the studying It convenient is effects to these introduce Conclusion the experience will coil two-turn of the above heating Curie point,When the turn each Introduction to Electromagnetic End and Edge Effects and End to Electromagnetic Introduction Slab . Depending on the application on the if notproperly, specifics, . Depending addressed magnetic Y– Y– X Z Re Y ctangula coil Z r X H Centra part b l d el ec tromag Long netic endeffec itudinal 0 P/P Handbook of Induction Heating ofInduction Handbook c el ec Tr ed t tromag ansversa ge eff a ec X neti t l Z c Industrial Heating Industrial - - - - , Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 cylinder. of asolid length the (b) along distribution power density (a) heater normalized of induction and Sketch FIGURE 3.35 tioned above, electromagnetic end effect represents distortion of the electromagnetic field electromagnetic of the distortion represents above,tioned end effect electromagnetic 3.35 men Figure As shows solid when cylinders. heating of appearance end effect the 3.1.7.1 we phenomena. provide these abrief orientation regarding section, applications. this different In in edge end and effects electromagnetic of the tures coils when billets are transferred transversally in respect to the coil. to the respect in transversally transferred are when billets coils oval-type coincide. in However, role billets when play heating can essential an effect this cylinder the and the coil of fluxsolenoid inductorsaxes the of symmetry gitudinal when workpieces (e.g., negligible cylindrical when heating typically is or bars) lon the billets in tribution across the slab width the (along across tribution the cross section, Figure 3.34). Figure section, cross workpiece (along or Z rectangular the cylindrical of the length along the interest. of engineering great is effects For edge applications,end practical and many effects. separate study the of edge end and the of both of result a mixture the is heat distribution source corresponding the and 3-D (3-D) zone of where field three-edge excluding the corners only is the space distribution plate. infinite Basically,the fieldin to the corresponds tion have and effects edgeend a 2-D fielddistribu electromagnetic part, the central the In area. end effect a longitudinal and Background Theoretical Appropriate sections of Chapters 4 and 6 provide a detailed analysis of the main fea main ofAppropriate the of 6provide Chapters 4and analysis adetailed sections The analysis of the edge effect is often conducted often is by edge of evaluation the effect analysis power ofThe the dis density power the distribution It density convenient is by estimating to study end effect the Electromagnetic Longitudinal Effect End (b (a) )

Normalized σ 0.25 power density 1.0 P/ c R ab in thecentralpar “P Ex c ” issurfacep F Central (reg F F treme end 1 2 3 Induction coil S X olid c -axis in the the in -axis Fr F 1 e >F quencie ylinder ular) pa 2 ower density >F t ofc 3 s: ylinder rt Cold endofc Y – cross section). X cross edge The effect F 3 Tr ansition zone ylinder b F 1 -axis in the Y the in -axis Z Z – 79 Z - - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 nounced skin effect and constant current. The presence of the large coil overhangs coil large of presence the ( The current. constant and effect skin nounced pro solenoid of inside amultiturn inductor assuming and positioning different end takes (left) when bar its trailing cylinder nonmagnetic of the length along the distribution sity heated compared to its regular region (Figure 3.36b). (Figure region heated compared to its regular overhang, coil pipe of end region noticeably the the is of substantial under regardless contrast, SAEIn an when 1018 heating (60 frequency Hz) line pipe steel carbon and using severe 3.36a). overheating in end (Figure itself tube of the manifests end effect magnetic overhang, coil electro alarge having an and 30kHz using acopper of tube heating case hollow the statically when appearance In heating of cylinders. end effect cases extreme overheating workpiece. end of the 3.36 extreme Figure the shows illustration, an As two orunderheating overheating (“hot”) extreme the end. For a single-turn coil, K For asingle-turn width)/(pitchturn coils, its For value 1. winding). always multiturn is than less of turns uniform power density distribution. In static heating, small overhangs application and small static heating, In power distribution. density uniform properties of the heated ( of the material properties physical electromagnetic the Fand frequency of the effects The depth. rent penetration is defined primarily by the following by variables: defined primarily is end (“hot” at extreme the end effect end) 3.35, (Figure magnetic cylinder of the “a”– region) of conventionally case cylinder. end electro of coils, the the the designed extreme In the in 80 σ wound. are K pitch) coil (also to as referred how indicates turns the tightly case shown in Figure 3.37, Figure shown in case it approximately to σ corresponds overhang coil (in the acertain is there time, At comparedinder part. to same its central the the pipe is noticeably underheated. noticeably pipe is the of end the overhang, coil of substantial (60 regardless Hz) and frequency line 1018 pipe using AISI steel carbon heating (b) When kHz. 30 using tube acopper IH when end tube of the overheating severe in itself manifests effect end (a) electromagnetic An cylinders. hollow heating when appearance effect of end Two cases extreme FIGURE 3.36 where 7 ) results in substantial power density surplus in the end region of the nonmagnetic cyl nonmagnetic of end region the power the in surplus density substantial in ) results As an example, an As 3.37 Figure overhang coil of shows the power σon surface effect the den abovementioned of combination the lead factors incorrect to or underheating can An • • • • • • Skin effect, R/ effect, Skin Ratio R Ratio Coil overhang, Space factor of coil turns K Space factor of turns coil of flux Presence concentrator Power density is the radius heated of the the R is cylinder, R i /R (a) δ σ space = 1. An incorrect combination of these factors can lead factors either can to of combination these =1. incorrect An space ρ and and (density of windings of turns), (density coil of windings (b μ ) r ) are included into the skin effect ratio R/ effect skin included the into ) are i is the inside coil radius, coil inside the and is Handbook of Induction Heating ofInduction Handbook 4 ) that results in areasonably in results ) that space δ is the cur the is =(copper δ . σ K 5 , space σ 6 ------,

Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 axis with “ with axis loop produces empty aBfieldalongthe loop Acurrent-carrying woundN tightly turns. conditions: following “ideal” of an assumption loop The asingle of wire. solenoid in tribution the presumes its dis describes expression that an solenoid relatively using be can coil obtained easily example. simplified 3.35, (Figure following part the central using zone “b”). statement illustrated be can This workpiece the opposite coil cold in that the under end (the end) area half only is so-called the cylinder, induced of in eddy the density homogeneousa the nonmagnetic current with bar. end of the extreme noticeableof relatively with deficit associated are low of power frequencies the at density Background Theoretical processing through a multiturn coil. amultiturn through processing during cylinder of anonmagnetic end left of the positions at different effect end longitudinal of the Illustration FIGURE 3.37 Figure 3.38Figure “ideal” shows of an sketch such the solenoid of length theempty area endtheof “ideal”in fieldaxis alongthe magnetic of the distribution The coil long multiturn electromagnetically It of an possible case is to the show in that • • • solenoid. of the proximity conductive located no within bodies electrically are There negligible). is (the effect skin turn each within distributed uniformly is current coil The wire. thin wound tightly are using solenoidThe turns z ” component following expression [1,44,60–65,83]: to according the Po of trailing sitioning end Surface power density (%) 100 120 140 160 20 40 60 80 –100 0 σ σ 1 σ 1

= –80mm,

1 = –80mm σ 2 –20 σ σ 3 5

= 90mm, σ 5 σ

σ = 90mm 4 2

= –40mm, 60 M σ 5 ultiturn coi Leng σ 6

= 170mm, th (mm 140 σ σ 3 6 l

= 0mm, ) 220 σ 7

σ = 210mm, 7 σ 4

= 30mm, coil Mu He 300 at ltitur and radiusl and ed bar n R that has 81 - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 Sketch of a tightly wound multiturn solenoid for end effect calculation. effect end for solenoid multiturn wound of atightly Sketch FIGURE 3.38 Equation 3.13 in resulting as written be field along thecenter can coil magnetic the length, coil the 82 expansion of B expansion μ where the contributions of all current-carrying sections. Therefore, after integrating d integrating Therefore, after sections. current-carrying of all contributions the will be “d section current-carrying contribution of asmall the of wound tightly assumption turns, 0 The magnetic field at the center of the empty loop can be obtained by assuming assuming by fieldbe obtained the magnetic centerthecan empty loop The at of The magnetic field distribution along the axis of an empty solenoid can be obtained by solenoid empty be obtained an of can axis alongthe fielddistribution magnetic The The total magnetic field in the center of the coil can be obtained by taking into account taking by be obtained thecan centerthe field coil in of magnetic total The After simple mathematical operations, the total axial field in the centersolenoidthe field in of simple axial operations, total mathematical After the is the permeability of free space of (a free permeability vacuum), the is Z ” on the total field in the center of the solenoid will be as follows:thebe centersolenoidthe field in of total will ” on the is the loop current, and and loop of current, the interest, loop area Iis to the the from distance axial the Z is z of a single wire loop on a multiturn coil. Taking consideration into loop on a multiturn the of wire asingle dd B z = 2 () RZ 22 µµ + 0 B symmetr R z Axis of 2 = l/2 32 µ // B R 0 B y z RN z 2 NI = 2 l = l B 2( z RZ I () = Z µ 4 µ 22 − L 0 ∫ L µ Rl 2 0 + = 2 RI 2 0 NI 22 R μ () 2 I + RZ 0 0 . =4 ) 22 (3.14) RN 32 2 / + d 2 l . Z π Z , (3.17) × 10 × (3.13) I l/2 32    / () −7 RZ . Handbook of Induction Heating ofInduction Handbook (3.16) H/m [or Wb/(A [or H/m 22 dZ + d Z 32    . (3.15) ∗ m)]. B = 0 in Z = 0 in z along along Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 magnetic flux density B flux density magnetic ter of that in the center ( the ter of in that to aquar equal power end is coil the middle that coil. the of under It the density means the in than less twotimes sufficientlyworkpiece end with coil long is the nonmagnetic workpiece. nonmagnetic homogeneous long infinitely an with coil long multiturn electromagnetically forroughly expanded an (being more typical). (being cylinder,netic overhangs, coil even large having may either be overheated or underheated cylinder, ferromag of ends the the nonmagnetic of the those there. Therefore,tion unlike cylinder). of anonmagnetic factor second apower end effect causes to The the reduc by factors two [1,9,44,76–81]: affected mainly piece is Equation 3.16, 3.13 Equations 3.16 and into transformed be can in long solenoid.netically limits It corresponding possible the is to show by that letting piece radius,” and the coil turn spacepiece radius,” factor K turn coil the and heated the workpiece, in effect ratio skin the of on the depends “coil radius inside to work thanks to thanks lines have flux one. materials to magnetic a tendency gather Magnetic the nonmagnetic gaps). air load the small (for and and coil the 12 effect gaps equivalent between pronounced air skin poor coil-to-workpiece is not there is pronounced and effect radiuscoil (if skin

long coil), it possible then is to neglect Background Theoretical as ten A comparison of 3.18 Equations A comparison 3.20 and shows coil, the empty at of ends that the the As discussed above (see 3.35, Figure discussed As zone “b”), under induced of current the density the The first factor causes an increase in power the increase at workpiece’san factor first causes The end (somewhat similar This is the well-known expression for the axial Bfield the centeran electromag well-knownat axial expression forof the the is This The electromagnetic end effect in a magnetic slab has several features compared to the compared to the several slab features has amagnetic in end effect electromagnetic The Thus, for an electromagnetically long coil, Equation 3.19Thus, electromagnetically for an approximated be can as (electro-magnetically R(electro-magnetically its radius solenoid of l>> the much length than greater is the If 2. 1. dency to gather the magnetic field within the magnetic magnetic theworkpiece. fieldwithin magnetic to gatherdency the have which volumetric and surface currents, a often the effect magnetizing The out workpiece. of the field tend to that force magnetic of eddy the effect currents demagnetizing The μ r . Generally speaking, the electromagnetic end effect in a ferromagnetic work aferromagnetic in end effect electromagnetic the speaking, . Generally z drops to one-half its value to one-half drops at center. conclusion be the can This P end = 0.25 = ∗ B P z center with respect to respect R with B B = z z 2( = = ). of zone factors, “b” Among other length the µ µ µ Rl 0 0 0 2 space l 22 NI NI NI l + and might be as long as four times the the long as times four be might as and . . (3.20) (3.18) ) .

and Equationl and 3.16 rewrit be can coupling) or (3.19) - 83 ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 bar end heating using a solenoid multiturn coil with tighter wound turns and a“cold” and wound tighter with asolenoid turns coil using end multiturn heating bar modeling. computer numerical after application obtained be can related to aparticular subtleties its and effect of this accurate controlled. be estimation An can effect or howance this its appear factors affect that of abasic providing only some critical illustration end effect 84 Material is AISI 1039. AISI 1kHz. is is Material Frequency heating. end bar in time cycle versus distribution temperature of surface simulation computer of FEA Results FIGURE 3.40 bar. steel carbon a heating end when (top) mesh distribution FEA field acoil and FIGURE 3.39 Case StudyCase of electromagnetic introduction aboveto a subject asimplified as serves discussion The 1000 Temperature, °C 1200 . As an example, an 3.39. As Figures 3.40 and of show simulation FEA of results the 200 400 600 800 0 05 1 kHz 0 100 Bar length,mm 1502 Handbook of Induction Heating ofInduction Handbook Heat time 10 s 15 s 21 s 27 s 35 s 45 s 5 s 00 25 0 - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 slab’s end areas, similar distortion occurs at its 3.34).slab’s edges (Figure occurs phenomenon distortion takes This similar end areas, workpiece,the fieldin magnetic of the distortion besides the a rectangular heating When 3.1.7.3 windings. of turn accordance helix to the in pucks twisted forces coil induction of the coil, induction thick). magnetic 6mm Upon and amultiturn (36 entering diameter mm example,pieces. an As 3.43b Figure of puck-shaped shows realignment workpieces brass in end effect. in avariation with associated is that workpiece along the circumference nonuniformity ture turn’s of the reduceon tempera effect the helix or dramatically help eliminate will which straight-wounded with should designs coil made be Efforts to use effect. turns, coil helix shows of presence of appearance the ahot the the spot at pipe the end illustrating area, of variation power density. more pronounced in circumferential result will winding pitches large and of wide turn copper using tubing frequencies, Higher seen. pipe the (top end region within induced vs. heat bottom) distribution clearly be source can with field and,it,compared toassociated its Variation bottom. electromagnetic the in design. coil aconventional and Hz of multiturn 500 afrequency pipe steel bon using car × mm diameter mm wall ends 19 of 250 field when heating distribution tromagnetic example,an 3.42 Figure shows of elec computer results the the demonstrating modeling As windings. of turn ahelix with cold conventionally coils using spots wound multiturn hot and responsible of appearance for localized an also is end effect electromagnetic The Effect Helix 3.1.7.2 (a and b, respectively). 3.41 Figure in lower using frequencies illustrated higher is and cases these in end effect relatively and coil-to-workpiececoils small gaps (e.g., 2–5 mm). basic of The appearance material. of anonmagnetic end effect the represents end effect Finally, bar the Curie point and the exceeds end temperature bar the stage. this during occurs bodies nonmagnetic and of magnetic ofcombination end effects below Curie temperatures point. the A complexwith properties ferromagnetic their retain However, properties. of loss magnetic the with associated is that regions subsurface the place. At stage, that layers surface heated the are above bar of the Curie temperature the body. of stage aferromagnetic heating end takes effect the as interim itself With an time, 3.40. shown on Figure stages are heating at different inductionthe ends coil. of both near field complexitydistribution the magnetic of the heated abovecation regions bar of the and below and Curie point. the Note end effects the 3.39. Figure shown in are clearlyindicates a demar lines flux Aconcentration of magnetic temperature of to afinal bar of1160°C the ± 35°C. 210 mm. to heat it 45 sof a100-mm-long end at heating, was left required After the section SAE was bar steel is of the end. material 1039. The length total The 1kHz. is frequency The Background Theoretical The helix effect can also manifest itself by unanticipated movement by itself unanticipated of heated manifest work also can effect helix The theof results computer confirm modeling. For example, trials Heating 3.43aFigure noticeably is there overhang greater at top coil of winding, the coil of ahelix Because short applying end effect electromagnetically an with associated are Several subtleties manifests stage, end heating effect and whole the magnetic is bar initial an During endandthe profiles of effects temperature of surface appearance dynamics Sequential stage heating final computer-simulated the mesh and FEA fieldat distribution magnetic Electromagnetic Transverse Edge Effect 85 - - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 86 Coil field distributions attributed to a helix effect. helix to a attributed distributions field Coil FIGURE 3.42 (b). quency (a) fre high versus low frequency applying coil asingle-turn in effect end of electromagnetic Illustration FIGURE 3.41 (a) curr Eddy Pa en rt t Coil curren Single- coil tu t rn (b ) curr Eddy Pa en t rt Coil curren Handbook of Induction Heating ofInduction Handbook Single- coil tu t rn - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 highly pronounced skin effect ( effect pronounced skin highly withthe a slab’sin section field intensity cross electric oftransverse the distribution the flowthe contourthematches induced current closer slab.of the effect, 3.44Figure shows always is temperature located there). maximum the that more skin The pronounced the slab’s of located surface the is on the rent density (it part central not, does however, mean ing the required temperature distribution across the slab or the plate across width. distribution temperature required the ing plays that edge effect amajor obtain role transverse in place electromagnetic to the owing Background Theoretical Distribution of the electric field intensity ( intensity field electric of the Distribution FIGURE 3.44 Industrial Heating 2, Industrial Part magic,” “black is no longer strips and bars of slabs, heating V.induction flux Rudnev, D. Longitudinal Loveless, University, 1986; Russia, and Engineering El. Technology, St. Petersburg of Electrical Department Thesis, PhD slabs, and cylinders of large-dimensional heating of induction control optimal and simulation Mathematical The helix effect manifests itself in the hot spot at the pipe end area (a) and by realigning pucks (b). pucks (a) area pipe end by realigning at and the spot hot the in itself manifests effect helix The FIGURE 3.43 to not 10) is pronounced ( effect skin when and the edge area often occurs. often edge area quite pronounced, is the heat the in effect surplus or copper slabs skin whenminum, the sides (two- side).alu three the and from penetrate surfaces steel, magnetic heating When sides two from (from opposite two wide surfaces), but at heat edge the the areas, sources heat penetrate part, the sources central the in because occurs This part. central to the pared overheated easily be can edge part, heat the areas at central com losses the than higher area is usually (1.5–4.0)* slab (2-D along perimeter, edge the same the areas except in 3.44). corners, Figure edge The of the slab and most of the induced currents close their loops earlier, their the close without most slabof induced of and reaching the currents the contour not does the match section slab current’s cross case, the the pathplace. in this In When heating a homogeneous nonmagnetic body, ahomogeneous value heating nonmagnetic When eddy of the maximum cur the If the skin effect is not distinct ( not is distinct effect skin the If pronounced ( is effect skin the If Central pa Corner area Edge area He (a) loss , February, 46–50, 1995.) 46–50, , February, at rt δ long. at are edge losses the area surface Even thermal though He d / at lo δ d d =10, d where slab thickness the / / E ss < 3), then underheating of the edge areas will take δ <3), take will edge of underheating the areas then ) in the transverse cross section of the slab. V. (From of the section Rudnev, cross transverse the ) in > 5), then the current density is approximately is density δ >5), the current the then Y 0% b (b ) 20% d/δ =10(pronounc d/δ =3(poor“s 40% Ey d 60% / δ Ex = 3). = 80% E 100% ki n” d ed eff “s ec ki X t) n” divided by δ by divided eff ec t) is equal 87 - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 ties (Figure 3.46b). (Figure ties #1 3.47 billet Figure for shows within power the distribution density but workpieceswhen resistivi have both nonmagnetic are noticeably electrical different (EEJ effect).erties prop different with materials of joined effect electromagnetic the phenomenon called is #2), (billet cylinder This power surface the magnetic of decreases. sharply the density #1), (billet cylinder nonmagnetic At end left power the the increases. sharply density billet’s (the joint area). billets so-called the At between right end of area tion the the and quitetransi complicated the fielddistorted is is where magnetic the area Another 3.46a). (Figure bodies magnetic and nonmagnetic of the end effects to the attributed power #2), distribution anonuniform density be (billet billet will there magnetic of the 3.46a). (Figure rather be different will billets nonmagnetic and netic power surface the time, mag At of densities same the the current. coil tocorrespond the be and approximately same thewill areas central their fieldintensity at magnetic the then long are enough, 3.46. billets Figure both and shown coil induction in is the If case this in solenoid amultiturn inductor. located power surface ofare in the density distribution The properties) its magnetic maintained billet other the and nonmagnetic became point and physical (e.g., different properties substantially was heated one billet above Curie the distorted becomes two[1,77,85].the billets field coil, induction the between an inside or placed joined other to each are close properties material different with billets two When tion coil with two cylindrical billets (Figure 3.45). (Figure billets cylindrical two with coil tion aconventional in solenoid process induc consider electromagnetic let us effect, the of this field. magnetic acommon located are in the study metals whenTo different two simplify (EEJ effect) properties occurs different with materials of joined effect electromagnetic The 3.1.7.4 application the [1,44,79,80,84]. design of adual frequency requiring part central the stage heating couldbe 20% final lower the in the temperature of compared to areas corner superalloy slabs or RCS relatively large using bars low frequency, of temperature the the steel, or Ni-based stainless nonmagnetic slab.the titanium, For example, thick heating in of part central the values in corresponding the than less be will edge areas in densities 3.44). (Figure even edge the areas 2-D sometimes and area corner aresult, power As the 88 Sketch of an IH system to study the electromagnetic effect of joined materials. of joined effect electromagnetic the study to system IH of an Sketch FIGURE 3.45 When discussing the EEJ effect, it is necessary to mention that this effect also occurs occurs also effect to mention this that EEJ it the effect, necessary is discussing When #1, (billet billet 3.45) Figure At nonmagnetic end of left the at right end the and the let’sFor purposes, with illustration what review billets steel when carbon happens two Assume that the billets have different electromagnetic properties (e.g., properties have electromagnetic billets different the that μ ρor Assume different Electromagnetic Properties (EEJ Effect) (EEJ Properties Electromagnetic of Materials Joined Electromagnetic Effect with Different 0.127 m 0.1 m C A Billet #1;ρ1 0.1 m(4″) 254 mm(10 B D Induction coi Billet #2;ρ2 ″ ) 0.1 m(4″) Handbook of Induction Heating ofInduction Handbook l r ). ). - - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 Theoretical Background Theoretical vs. the opposite case (b). case opposite vs. the (a) #2 billet of the #1 threefold is billet ρof the when of EEJ effect appearance different 3.46). substantially Note ρ=1.1 Hz; =60 #1 (frequency μΩ of billet length the along distribution Power density FIGURE 3.47 when heating nonmagnetic billets having different electrical resistivities (b). resistivities electrical different having billets nonmagnetic heating when (a) and materials vs. nonmagnetic magnetic heating (EEJ). When materials of joined effect Electromagnetic FIGURE 3.46 1.2 1.6 0.4 0.8 (a) 0 ρ 1 C =3×ρ Lengt 1 h, in. 2 P ower densityratio, 2 AA (b (a) ) 3 P

0 Surface power density

P/ Surface power density 4 c Nonmag D 1 bille 1.4 B µ Ra 1 1.8 , ρ t 0 netic 0.4 0.8 dius 1.2 1.6 1 µ , in. 1 ρ

= µ 1 Leng Leng

> ρ 2

= 1 2 th th 1.6 1.2 0.4 0.8 (b) 0 Tr Mag ρ 1 (EEJ effe ansition are bille Lengt =0.33×ρ µ Tr C 2 neti (EEJ eff ansition are , ρ t 1 2 h, in. c ct ec P ) 2 ower densityratio, t) 2 a a 3 Z Z Z Z P , compare with Figure Figure with , compare 4 0 P/ D c 1 B 1.4 Ra 1.8 0 dius 0.4 0.8 1.2 1.6 , in. 89 Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 temperature. Alloying and residual have and elements can temperature. ameasurable Alloying impact on k 3.48Figure [71,72]. of function anonlinear conductivity one is may As note, thermal the general rule. this from exceptions be ( temperature of the afunction also is which ( workpiece lead excessive to an also distortion. can the of amount needed in heated mass than A larger residual and stresses. characteristics microstructural negativelysomecan and, efficientin cases, energy affect less process ing adjacent the power workpiece of the in mak areas consumption, increases increase ture to hardened, be but adjacent is not. tempera well,region, which are as in The which areas place the take not in only will heat rise temperature transfer, of aresult the As intense workpiece. the within distribution temperature the promote equalize and heat transfer of shafts),ing value adisadvantage ahigh of quite of often its kis because to tendency 3.47a).(Figure ρ When (i.e., resistivity electrical #1) right endhigher of with the billet billet of the area reduced is 3.46b), (Figure #2 of that billet times joint power the surface three the in is density then ( resistivities electrical different but 3.45. have nonmagnetic, Figure they induction system are shown the in billets Both 90 heating applications. heating workpiece, through the in important is within which distribution temperature a uniform heat [73]. losses Conversely, it easier to high, obtain is heated is kof when the the material and lower efficiency surface thermal tolower higher corresponding value required of kis or unfavorable. For inductor’s for example, an amaterial choosing or aliner, refractory a alowwith k ductive workpiece. conduct amaterial kvalue heat ahigh will with faster than Amaterial rate conductivity the con kdesignates at heat which travelsThermal athermally across 3.2.1.1 3.2.1 3.2 above Curie point. the workpiece, tothe heat but whenjust relatively long it required billets is especially thin within distribution temperature may have final appreciable and an transient impact on the edge end however, and effects; netic some phenomenon applications are there where this (Figure 3.47b). (Figure κ ) and the electrical conductivity ( electrical the ) and The values of the thermal conductivity of some commonly used metals are shown in shown in are metals conductivity of used some commonly valuesThe thermal of the The Wiedermann–Franz law governs the relationship between the thermal conductivity thermal the law between relationship governs the Wiedermann–Franz The However, applications heating selective (e.g., in harden or case hardening surface gear The EEJ effect does not usually play as important a role as skin effect and electromag and effect play arole not does skin usually EEJ as important The effect as Thermal Phenomena IH in Thermal Thermal Properties of the Materials of the Properties Thermal hra Conductivity Thermal . Depending on application specifics, a particular value on application be beneficialof kcan . Depending a specifics, particular 1 =0.33 ×ρ ρ 1 σ and ρ and 2 ) for the majority of pure metals and metallic materials, materials, metallic and of metals majority ) for pure the , there is an increase in surface power surface there in density increase an is , there 2 ). When the electrical resistivity of billet #1 of billet ( resistivity electrical ). the When T ). However, some alloys (e.g., irons) cast may Handbook of Induction Heating ofInduction Handbook . ρ 1 - - - ) - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 speaking, Mathematically change. workpieceby temperature the of required to achieve aunit valueThe of heat would that of amount capacity energy the have Cindicates to absorbed be 3.2.1.2 Background Theoretical is measured in J/(mol in measured C is °C). where d Thermal conductivities for some metals versus temperature. versus metals some for conductivities Thermal FIGURE 3.48 plain carbon steels SAE 1010, 1042, and stainless steel 304. SAE 1010, steel steels carbon 1042, stainless plain and and pureAl, Cu, for W; for (b) temperature heat vs. (a)temperature. vs. Specific metals some heat for Specific FIGURE 3.49 are shown in Figure 3.49 Figure shown in are [71,72]. valuesheat temperature.metals The specific used of tosome aunit the of commonly mass J/(kg °C) or Btu/(lb °F). in measured cis The increase. temperature to achieve aunit material of the mass by aunit to absorbed be energy required of amount the the heat meaning mass, capacity unit per Heat heat specific capacity closely related is called to a parameter A higher value of specific heat corresponds to the greater required power valuegreater to required theunit A higher heat heat of to specific a corresponds J/(lb. °C Heat Capacity and Specific HeatHeat Specific Capacity and is the required energy and d and energy required the Q is (a) 100 200 300 400 500

00 Thermal conductivity, ) W/(m °C) 100 200 300 400 0 400 Te mp erature, °C 400 Te 800 mp T Aluminum Copp ungsten erature, °C 800 er 1200 T Aluminum Copp J/(kg °C 1200 ungsten 880 220 440 660 1100 er is the required temperature change. Heat change. temperature capacity required the T is ) C 1600 = J/(lb. °C d d Q T 1000 1200 200 400 600 800 (b , ) (3.21) Thermal conductivity, ) W/(m °C) 10 20 30 40 50 60 04 200 Te mp Te St St St 00 400 ee ee ainless st mp erature, °C l 1042 l 1010 erature, °C c 8001 600 , which represents the the represents , which Ti St Carbon st ee ainless st tanium l 800 200 ee J/(kg °C ee l l 880 1320 1760 2200 2643 440 ) 91 Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 Source: Variation range 7075 7075 7072 7050 Pure aluminum 7005 where q 3.2.2 some exceptions. are of there Al, though pure lower those usually alloys are than of Al). that pure 93%is than greater Values heat specific conductivity and of of thermal (i.e., of aluminum pure those ρof alloy than 7075-T6 greater typically alloys are aluminum alloys at (20°C). temperature room of aluminum resistivities some 7000-series Electrical example,factors. an As Table 3.7 shows appreciable an of variation physical of properties residual size, plastic elements, deformation, prior grain heat some treatment, other and ductivity Fourier’s is tion law, toward low-temperature by the conduc heat basic transfer The law describes regions. that workpiece of the regions by high-temperature conduction the from Heat transferred is 3.2.2.1 [88–99]. ent pres convection, modes of radiation—are heat and transfer—conduction, three IH, all In Aluminum Alloy Aluminum 20°C Pure versus at Temperature, Alloys Ambient Aluminum 7000-Series of Some Variation of Physical Properties TABLE 3.7 92 in aworkpiece gradient (temperature temperature in to the proportional is difference) and versus pure Al versus of property ofalloy Similar to the electrical resistivity discussed in Section 3.1.1.1, Section in discussed resistivity electrical to the con values the Similar of thermal As one can see from Equation from 3.22, see one to according can As Fourier’s law, rate the of heat transfer Three Modes of Heat Transfer Modes Three hra Conduction Thermal 1979. ASM, MetalsHandbook,Vol. 2,PropertiesCleveland, andSelection:Nonferrous AlloysandPure Metals,ASM, cond are also affected by chemical composition, of presence by the chemical affected heat also specific care k and is heat is conductivity, flux by conduction, thermal kis temperature. Tis and Resistivity +7%; +93% Electrical Electrical μΩ 0.043 0.052 0.029 0.037 0.027 0.049 0.04 *m qk co nd Conductivity −38%; +7.6% W/(m* =− Thermal Thermal 155 130 227 180 137 166 211 gr ° C) ad () T ,

Specific Heat Specific −8.4%; +3% J/(kg* Handbook of Induction Heating ofInduction Handbook 960 893 860 855 933 ° C) Temper Treatment Commercial grade T73 T73 T6 T6 O O (3.22) - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 free convection q losses free heated workpiece. stationary of the losses to forced convection (e.g., much higher are convection 5–10 free often higher) than times travels strip at up the tions, to aspeed 5m/s (16 ft/s). Therefore, heat attributed the losses applica convection. coating some strip consideredIn free be as cannot heattion transfer of applications IH (e.g., gears, or disks, shafts), of rotating strips, heating wires, convec face or forced conditions, convection. anumber whether it free and in is Remember that at high speed (e.g., speed at high disk). of astrip, heat induction treating or wire, spinning (i.e., velocity or workpiece of the quenchants), the viscosity it if rotates their or moves and or fluids gas, air surrounding of workpiece, of the the properties properties thermal the respectively. and and

where where face heat transfer coefficientface[W/(m heat transfer ambient area, the and workpiece the between surface difference temperature to the proportional rate directly is Newton’s convection law heat states heatthe that transfer describe law transfer. This can (i.e.,or air heated of surface the workpiece the from ambient area). to the well-known The conduction, out by by convection heat fluid, contrast carried to thermal is In transfer gas, 3.2.2.2 temperatures. different with regions between distance to the portional toward cold the core. Conversely, rate the by of conduction heat inversely transfer is pro hot workpiece of surface the value the from intensivea high heat of in kresult transfer core (which, and surface between e.g., and hardening) place surface during takes typically conductivity workpiece. of the words, thermal other the In gradients temperature large Background Theoretical where the surface heat transfer coefficient represents the cooling intensity. the cooling coefficientrepresents heatwhere surface the transfer heated to 600°C (1112°F) atmosphere ( surrounding the into Celsius. Fordegrees example, convection workpiece the heat free from density loss surface There are several empirical formulas that provide a rough engineering estimate of the of the estimate provide that formulas several empirical are arough engineering There Value of convection losses can vary dramatically depending on the temperatures, sur Value on the depending dramatically vary of convection can losses thermal the of function a is convection coefficient primarily The heat surface transfer Convection heat transfer plays a particularly important role in the quenching process process quenching the roleConvection plays in important heat transfer aparticularly T a is ambient temperature. The subscripts “s” ambient subscripts temperature. is The “a” and denote ambient, and surface q T conv Convection of the Mode Heat Transfer s and T and is heat is flux by convectiondensity (W/m a are surface temperature and ambient temperature, and temperature surface correspondingly, are in conv , including Equations 3.24 Equations , including [100] 3.25 and [11] heating for billet qT qT co co nv nv =− 2 =− .°C) or W/(in. 15 qT 18 .( co .( nv 4 6 =− sa sa α () T T sa )[ )[ 13 13 . 2 . T .°F)], 32 W/ W/ , 2 (3.23) or / . n W/i r o T m m s is surface temperature (°C temperature surface is or °F), T 2 a ] ], =20°C) Table shown in as is 3.8. (3.24)

2 ), ), is the convection the sur α is (3.25) 93 - - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 sivity of the metal and σ and metal sivity of the of the free radiation losses q radiation losses free of the estimate provides to that aformula convection is losses, arough there engineering Similar heated to 1250°C (2282°F) atmosphere ( surrounding the into

radiation loss coefficient C coefficient loss radiation rate heat radiation, states by to the a that transfer which radiation proportional is thermal law of propagation Stefan–Boltzmann The energy difference. to a temperature attributed aphenomenon as by introduced of be radiation heat of can effect electromagnetic transfer (vacuum). region anonmaterial hot including workpiece the from The surroundings into radiation, mode heat the of may heat transferred thermal be third transfer,is the which In 3.2.2.3 94 Nonpolished, oxidized Polished Condition Surface Metals Nonpolished versus Polished Metals Used Commonly εof Some of Emissivities Comparison TABLE 3.9

roundings. However,roundings. some applications are be there where radiation can the heat transfer heat heated radiationworkpiece the sur free body from is the into when there geometry where T polished versus nonpolished metals is shown in Table shown in is versus nonpolishedpolished metals 3.9. of used some commonly of emissivity Acomparison nonpolished metal. than roundings radiates heat oxidation. surface less metal sur to polished in the time, At same the increase factors), an conditions. with surface For and example, increases value the of emissivity is shown in Figure 3.50a. shown in εis of and temperature afunction radiation as density loss Thermal For example, heat radiation flux radiation)density (free steel ( slab carbon of a The above-described calculation of radiation heat loss is a valid assumption for classical of radiation for calculation heat classical avalid assumption above-described is loss The The radiationThe heatbe approximated coefficient loss as C can radiationThe heatincludes coefficient loss emissivity, radiation shape factors view (the s Radiation of the Mode Heat Transfer and T and q ra d q 3.17 Equation to According Density Loss Convection Free of Calculating Examples TABLE 3.8 or q conv =× a conv are measured in degrees Celsius. degrees in measured are =1.86×580 56 .. =7.3kW/m 71 qT ra ds 00 Aluminum 0.042–0.053 − s =× 0.082–0.40 is the Stefan–Boltzmann constant [ constant Stefan–Boltzmann the is 84 1.3 56 s ×× and the value the and of =7.3×10 .[ rad 2 71 8 , () 0 1523 − 3 84 W/m ε () −= Carbon Steel Carbon 2 +− 293 0.71–0.8 273 0.062 TT 43 sa 44 or q q − conv 244 conv =1.54×580 () According to Equation 3.18 Equation to According T , governs this phenomenon. , governs this a =7.45kW/m ×= + 10 273 W/ 0.026–0.042 42 Handbook of Induction Heating ofInduction Handbook T 0.24–0.65 1.33 σ ][ Copper a s = 20°C) can be calculated as =20°C) calculated be can W/ =5.67 ×10 =7.45×10 m s =σ 2 2 m s 244 ε is the emis , where the εis ] (3.26) 3 kkW W/m −8 W/(m /m Brass and Zinc and Brass 2 0.03–0.039 0.21–0.50 2 . 2 K ε =0.8) 4 )]. - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 piece power ( average work required of for the value aballpark estimate used be heat of specific ccan workpiece of an the mass increase, by absorbed be temperature aunit to achieve aunit to required energy value thermal of the amount Since the heat of the specific crepresents 3.3.1 3.3 suitabletemperatures for hot 3.50b). working (Figure etc.) tungsten, to heat of workpiece steels,heating when titanium, the it required is to the high-temperature in applications heat the total of losses part (e.g., a significant are losses heated whention the workpiece or fast. moving rotating is of forced convec cases in lower approximatelytemperature than particular 350°C) in and low-temperature applications IH (i.e., at steel and lead, a magnesium, zinc, tin, aluminum, heater. heat of in shows major the part the losses convection analysis are that The losses valuethe of heat valueinduction loss. Ahigh the of heatof efficiency reduces total loss the several [88–99]. references found in be modes can of heat transfer complicated asimple such and approach would not valid. be three Complete of all details (kW/m density (a) loss radiation of the Variation FIGURE 3.50 convection and radiation heat losses. radiation and convection production rate. purpose, for Equation this 3.27 used be can Theoretical Background Theoretical Since thermal radiation losses are proportional to the fourth power fourth of to the temperature, proportional these are radiation losses thermal Since previously, discussed As reflects by convection heat radiation the and transfer typically Estimation of the Required Power and Dynamics of IH Power Required of the Dynamics and Estimation (a) Estimation of the Required Power Required of the Estimation for IH Radiation heat loss, kW/m2 100 125 150 175 200 225 250 25 50 75 0 0 P w 200 ) to heat agiven workpiece average to an at required the rise temperature 400 Te mp 600 ε =0.2 ε =0.3 ε =0.5 ε =0.8 erature, °C 800 1000 1200 Pm w = 1400 2 ) versus temperature and emissivity. (b) A comparison of (b) Acomparison emissivity. and temperature ) versus c TT fi (b He − loss t ) at n 1003 2126 , q q radiation conv (3.27) ec tion =Cs*(T =α*(T Te 50 62 mp s 4 s –T –T eratur a a 4 ) ) 500 932 e T, °C T, °F 95 - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 induced within the workpiece the P induced within (20°C) temperature room of to atemperature 620°C 120 in s, to be power the required (s). heat time required the is the billet length ( length billet the l is

Heat content of metals at various temperatures. at various of metals Heat content FIGURE 3.51 where of the material to determine the value the of P to determine material of the where 96 [J/(kg °C)], Equation 3.28 to determine the required power ( required Equation the 3.28 to determine as case, Equation 3.27 rewritten be can would approximately be to 70 equal kW ×hour/t. heat on the contentbased value. From 3.51, Figure value heat of the required content the range 20 toperature 620°C. Therefore, applying Equation 3.27, be power will required the For example, order in to heat asolid (0.1 copper cylinder mdiameter, 0.3 mlong) from In this case, the mass of the heated body can be calculated as heated of calculated be the body mass can case, the this In In engineering calculations, some value practitioners the prefer calculations, heat ofto use the content engineering In Figure 3.51Figure shows with values the heat used of the content metals for used commonly c =420 J/(kg average an as used °C) be value can heatthe tem specific of in the of copper is the density (kg/m density the γ is m is the mass of the heated of the body mass (kg), the is T in and T and Pm w = m m f are average values of initial and final temperatures temperatures average are (°C), final and and t values of initial ). c

Heat content, kW*h/t == 200 250 300 350 400 100 150 TT 50 fi π 0 − 0 D 4 t 22 3 n ) (for copper, γ=8.91 ×10 l γ 200 =× w 21 can be determined using Equation 3.27. using determined be can 31 PH .. w 400 40 =× 420 × 4 C Te × w 1 600 . Heat kW in content hour/t. measured is this In mp 620 Pr ×× erature, °C 120 03 od − .. P Al 8001 20 uc w c ) to heat (example a copper billet above) is the average the is value heat of specific Mg 89 ti == on 44 11 3 0001 kg/m . ×= ,. (3.28) 100 02 3 Handbook of Induction Heating ofInduction Handbook Copp 2001 Silver Wk 3 Iron ), ), is the diameter (m), diameter the D is and 1 er 44 kg 40 0 , 1 W . is is - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 where noid value coil, the of η etc.). conductive (i.e., bodies electrically other guides, and tooling, shunts,structures, fixtures, range of 0to 1.

ing after acontinuous caster). after ing uniform (e.g., are not temperature final induction of thecase slab/bar required and reheat temperature when initial reheating induction and hardening, selective hardening, surface as such cases in computationcal provide particularly can estimation much more precise

advantage of providing a quick estimate of the required poweradvantage ( required of the estimate of providing aquick have Such simplified formulas the required. is heating where through relatively uniform shaped workpieces of IH classically as such cations (e.g., bars, billets, slabs, blooms, etc.) and the workpiece the and power P cally conductive bodies located in the surrounding area area conductive surrounding cally the located in bodies as determined losses) be can and

where Background Theoretical (the so-called coil power). coil (the so-called Equation 3.29 power coil provides the P acorrelation between of P of As shown in Ref. shown long in As [6], sole electromagnetically an in a solid when cylinder heating The valueThe of The valueThe of It is important to remember that power that to remember It P important is Simplified formulas such as Equations as Equations and such 3.27 Simplified formulas are in convenient3.28 applivery use to w and electrical losses losses electrical and η P el lo el is electrical efficiency and η efficiency electrical is ss includes power loss in the coil turns includes turns power coil the in loss η P el lo su represents the ratio the power of represents the workpiece the induced in ss r also includes kW losses associated with undesirable of with includes support heating kW associated also losses el Pr PH can be roughly estimated according to the formula to according the roughly be estimated can w od = () w P η uc le : el el ss C ti = on . ××= 1 Pr + == od D D PP 21 120 lo η el 2 1 ′ ′ 1 ss uc th el P ρ ρ kg is thermal efficiency. thermal η is Both, =+ c = ti 2 1 s = on δ δ w PP lo tu 2 1 wl does not represent the power at coil terminals power not does the represent at terminals coil ηη ss rn 0 el P 0 P + = s . w . w 033 70 021 1 th os el P + P ×= , lo s lo tu su h (3.29) t D D ss 06 ss , rn r .. (3.30) 2 1 . = ′ ′ s 1

and kW and generated electri losses in 44 06 . µρ P ρ r 4 lo 1 su 48 t/ ss 2 r (as well as transmission kW (as transmission well as h , kW (3.32) . P w ). However, numeri el and η and P w th , to the total total the to , are in the the in are (3.31) 97 - - - - - c

Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 Equation 3.33 should applied: be cylinder. computer modeling; at the same time, there are several empirical formulas that can allow can that formulas severalcomputer empirical are there time, at same modeling; the diameter of acylinder,diameter made silver from or copper, η range of 0.8 the to in 0.95. usually is Curie temperature contrast, In billets when heating where 98 billet to water-cooledbillet guides). conduction (e.g., heat well to as as thermal vection attributed loss the heat from the losses η high notdoes the occur, then cancellation to alowcorresponds factor. current value that of this Therefore, assuming instead of using P of using instead of the coil and workpiece; and coil of the and and μ ρand in change on the depending varies cycle and heating the during workpiece ( resistivity, and coil have possible have the electrical and gap smallest high between the losses. An accurate estimation of the value of the accurate of the estimation An losses. stage) process applied. often is particular copper and cylinder (workpiece),copper cylinder and respectively; ρ as power determined be heating to can the pared and where When heating rectangular bodies, including slabs and plates, slabs and bodies, including of Equation instead 3.32, rectangular heating When The ratioThe following assumptions: the under Equation 3.32 obtained been has The next section shows power the workpiece next section that the The induced in The valueThe of η An application of thermal insulation or refractory can significantly reduce the heat significantly can or refractory application insulation An of thermal () • • • • P lo th An electromagnetically thick wall copper tube is used for fabrication. coil used is copper tube wall thick electromagnetically An field. magnetic inductor a single-layer,The is longsolenoid infinitely a producing homogeneous proximity. coil the in located conductive alone; no structures electrically are there standing is coil The pronounced. is effect Skin ss F D 1 includes heat losses from the workpiece the includes heat from to radiation con losses attributed surface and and and 1 ′ is an effective coil inside diameter, inside coil effective an is D () DD F 1 / 2 12 ′′ are effective perimeters of the coil opening and the heated the slab, and opening coil of respectively. the perimeters effective are D // th 2 in Equation 3.29 represents the amount of the thermal losses losses Equation 3.29 thermal of amount in the the represents

w → 1). For example, η the , the value, the of () ρµ 12 DD 22 ′ () =− r ρ el is typically in the range of 0.35 the to 0.45. in typically is el takes place workpieces when takes heating magnetic, are that P μ is referred to as coil electrical efficiency factor. efficiency η High electrical coil to as referred is δδ w av r 21 is the relative magnetic permeability of the heated of the relative permeability the is magnetic η ; (meaning the average the (meaning power cycle or per heating per el η th = and δ and = 1 el + PP when heating carbon steel cylinders below cylinders steel the carbon when heating w av F F 2 1 P 2 + 1 w are current penetration depths in the coil the in depths penetration current are av µρ () lo th ρ r P DD ss 1 1 lo and ρ and th 11 2 ′ . ss =+ (3.34) , (3.33) can be determined with numerical numerical with determined be can 2 are the electrical resistivities resistivities electrical the are δ Handbook of Induction Heating ofInduction Handbook 12 ; D ′ is an effective outside effective an is P w is not a constant not is a constant r . This is why, is . This () P lo th ss com el - -

Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 electromagnetic coupling and as a result leads to a decrease in η in leads aresult coupling to as adecrease and electromagnetic coil-to-workpiece larger oflation leads having that to anecessity an worsens gaps. This available having spacecantly. for requires its of instal refractory use the time, At same the shown [101]: as determined be can losses valuethe of thermal a refractory, as blocks concrete with coils For losses. of cylindrical those rough estimation Background Theoretical to a variation of electromagnetic and thermal properties of the heated of the workpiece, properties thermal and powerto of avariation electromagnetic factors, but not including by limited several nonlinear affected are IH of the dynamics The 3.3.2 or not. arefractory to whether to as decision use intelligent an make to-workpiece heat losses and more than compensate of loss for η the moreheat and than losses soldering applications. and brazing, is the inside diameter of the induction coil (cm), induction coil of the diameter inside the is and (dotted line). voltage constant is coil and (solid line) constant is current coil modes: process common most of the two for heat time of the afunction as workpiece the (b)within induced refractory. powerof of Variation thickness versus efficiencies thermal and (a) electrical Coil FIGURE 3.52 tory (cm). tory coil-to-workpiece gap maximizing η coil-to-workpiece gap maximizing and, therefore, at any not possible refractory all have cost effective to use and smallest the always is refractory areasonable it some efficient compromise. cases, more In is energy efficiency. it hand, other reducesthe Therefore, electrical a or use use not a to to decision

where ( η th An application of thermal insulation can improve can η application insulation An of thermal In other cases, it is advantageous to use a refractory and significantly decrease surface surface decrease it significantly cases, advantageous and other is In arefractory to use Totalinduction ( the coil of efficiency Therefore, on the one hand, refractory allows to improve the thermal efficiency, allowsTherefore, to improve one refractory hand, on the thermal the but on ) and electrical efficiency ( efficiency electrical ) and Intricacies of the Dynamics of IHIntricacies Dynamics of the () P lo

th Coil efficiency ss gap (i.e., extrusion). and helps modeling Numerical rolling, before IH forging, Ele is the heat loss from the workpiece heat the the from loss is (kW), surface (cm), length coil the lis D ctrical efficienc ickness ofrefractor ermal efficienc y η el P ) according to) according Equation 3.36 lo th y ss =× el 37 y . This is typical for majority of selective hardening, for of hardening, majority selective typical is . This . 41 η ηη ) is a combination of both coil thermal efficiency efficiency thermal coil of acombination both ) is = 0 − el 2 η lo th Workpiece power g . (3.36) 10 Cu D l    rr th D D 3 el ent isconstant is the inside diameter of the refrac of the diameter inside the is that is associated with greater coil- greater with associated is that and reduce heat and signifi the losses 3 1    , (3.35) He at time V el (Figure 3.52a). (Figure oltage isconstant 99 - - - 1

Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 δ of thermal conductivity k of thermal 3.54a, profile Figure temperature the power from the matchnot does because profile density stage. see one can As for workpiece this the typical is within differential temperature large of located a is existence at temperature surface. the Intensive the and heating maximum The 3.53. stage Figure study radius heating of workpiece shown ofthe in case the at initial an at core. the no practically change with temperature surface the to arapid in ing increase stage low. are layer induced surface power The fine the the of appears in workpiece lead relatively of the because at surface low this cylinder the temperature, heat from the losses ture (Figure 3.3). (Figure ture μ time, At same the line [318]. line inductionthe of of computer profilesalongthe temperature length of critical modeling production and 1kHz; rate is mm/s 65 is frequency (2.56 in./s). 3.53 Figure shows results 1 m (40 is of coil each length in.); 0.3 8; is is m (12 of number coils coils gap between in.); follows: as are parameters 152 is ID (6 (0.5 mm 12 is mm in.); in.); thickness refractory multicoil system. induction Coil in-line an (3-in.-diameter) using bar steel carbon medium Refs. [6,9];in provided be here. will thus, abrief discussion only found be modes study can of these Adetailed temperatures. ambientder to forging from Figure 3.52b shows of variation cylin steel power the acarbon when heating versus time cycle. heating entire the power coil during constant, and constant is is stant, current coil modes: process voltage coil classical con three is it possible mind, is to recognize in this long solenoid coil. Having electromagnetically located an inside cylinder steel a carbon conditionssimplified of heating IH considering of to evaluate beneficial is the dynamics supply control operational features, mode, system layout. and it For purposes, illustration 100 Industrial Heating Industrial slab, and of bar, rod re-heating induction in considerations temperature and Rudnev, al., D. et Efficiency Loveless V. (From 76 [3 is 1 kHz. mm in.]). is (diameter bars Frequency steel carbon of medium of IH dynamics Thermal FIGURE 3.53 2 is respectfully small, and therefore, the skin effect is pronounced. is time, At same effect the therefore, and skin the small, respectfully is During this stage, η the this During 3.54a shows powerFigure and temperature (heat the source) density along distribution At the initial stage of the heating cycle, the entire workpiece stage cycle, heating of entire the the magnetic, Atis initial the study a point case review a to of heating this beneficial 76-mm-diameter be at It will , June, 2000.)

Temperature, °C1000 1200 200 400 600 800 0 02 1 , which spreads the heat from the surface toward surface heat spreads the the , which the core. from el of the metal with tempera with metal ρof the in increase of an because increases Su 04 23 rfa Induction coil ce 06 Carb r remains relatively aslight reduction and of high, μ remains 08 Coil po on st Cor 45 ee e sitions l ba Ti 0 Av me, s r erage 1001 Refractor 67 20 y Cu 1401 Handbook of Induction Heating ofInduction Handbook rie poin 8 60 t 18 0 μ r is quite large, is - - - - r

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Industrial Heating slab, of bar, and Industrial rod re-heating induction in considerations temperature and stage, (c) (From stage. stage, 3.53. final (b) (a) intermittent V. Rudnev,Figure al., Initial et Efficiency Loveless D. on shown study case a for bar steel carbon IH a of stages different at profiles power density and Temperature FIGURE 3.54 does not affect the rise in in rise the notdoes affect Background Theoretical 2. 3. 1. layer μ and workpiece the in surface properties its magnetic loses steel carbon Second, the surface. the in heat in intensity decrease heat, of cleads peak to a the required to achieve the metal by absorbed be the 3.49).point (Figure value the must that Since of of amount cdenotes energy the value heat maximum has specific its the the ofFirst peak)(a all, near Curie increased thermal conduction heat flow thermal towardincreased a colder core. measurably, increased gradient is temperature in Surface-to-core resulting reduced. drastically be also (b (a) (c) )

Power density Power density Power density 0 0 0 After e After e After e r drops to 1. drops power aresult, surface As the (heat sources) density will 0.01 0.01 0.01 xiting coil#8 xiting coil#1 xiting coil#3 R R R adiu adiu adiu 0.02 0.02 0.02 η el s, s, s, . After a short time, coil electrical efficiency reaches maxi reaches its efficiency electrical coil a short time, . After m m m 0.03 0.03 0.03 0.04 0.04 0.04

Temperature, °C Temperature, °C Temperature, °C 1020 1220 1020 1220 1020 1220 220 420 620 820 220 420 620 820 220 420 620 820 20 20 0 0 0 After e After e After e 0.01 0.01 0.01 xiting coil#1 xiting coil#8 xiting coil#3 R R R adiu adiu adiu 0.02 0.02 0.02 s, s, s, m m m 0.03 0.03 0.03 , June, 2000.) 0.04 0.04 0.04 101 - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 intensity, frequency, ρ field magnetic the long solenoid of bodyinductor infinitely heatedfunction the inside as a consideration parameters. when process evaluating the where

of of value. approximately compared toA decrease to its threefold initial two- increased has steel stage). or intermittent transient the stage, called stage At ρ is that ing the this Curie point (second the exceeds temperature surface heat the after sometime cylinder of the 102 tems. The current production environment does not allow the luxury of process design of design process luxury not production does allow the environment current The tems. of sys design IH successful one the major of is factors the in modeling Mathematical 3.4 P 3.54c). (Figure distribution exponential classical have then disappear. The power will its finally density will pronounced less and becomes layer surface netic value the exceeds of 2* sources can be located in a subsurface layer asubsurface located be in can workpiece of the sources not and on its surface. value of heat conditions, maximum the to mention certain under here that It necessary is distance. at properties this its ferromagnetic retains steel place carbon the because takes This case), again. of approximately to increase particular it starts 1.4 (in surface this the from mm toward power core. surface. the the Then, However, decreases cylinder density at adistance at the heat of occurs the 3.54b sources Figure shows amaximum distribution. that nential expo steel. classical Power the from wave-like different aunique shape is has that density the layer than less is nonmagnetic of the place thickness stage the while takes This above to temperatures Curie point. materials the or magnetic any other steels carbon ing stage dual-property the when as heat characterized be cycle can heating of the portion This layers workpiece. ofMost the internal power of the the and surface now is the induced in lish its uniqueness. Some of those features are discussed in subsequent sections of this book. of this subsequent sections in discussed are features Some of those its uniqueness. lish andestab induction the process of the dynamics mightaffect that features specific has often asolenoid using inductor. applicationbasic phenomena were assuming Particular illustrated conducted computer by only numerical modeling. be can conditions heating including major parameters process the of all calculation Precise atity workpiece SI the to according system. the are frequency. surface, Fis units and All the surface of the workpiece, of surface the resistivity,the electrical μ ρis w Surface power density for ballpark estimation purposes can be calculated for a magnetic for amagnetic calculated be can Surface power purposes for density ballpark estimation Figure 3.54balongthe radius Figure shows profileand temperature distribution heat the source The existence of the three stages of IH results in variation of both the workpiece the of variation both stages in of power results IH three of the existence The Finally, the third stageFinally, nonmag of (a stage) the third place. the nonmagnetic takes thickness The workpieceThe however, nonmagnetic; becomes surface magnetic. layers remain its internal To finalize the discussion on the dynamics of IH, it should be stated that the above-discussed IH,of thetheit should thatabove-discussed be on stateddynamics discussion the To finalize and coil power coil and P μ r and an increase of ρ increase an and Advanced Induction Modeling Principles Mathematical and p 0 is surface power density measured in W/m power in surface is measured density c during the process of heating. All three stages should be taken into into stages should taken be three of process heating. All the during , and μ , and cause a 6- to 10-fold δ a6- cause in increase r according to according Equation 3.37 [6], pH 0 =× 27 ., 21 in hot steel, and the dual-property phenomenon dual-property hot steel, the δ in and 0 − 32 su rf ρµ 2 , H su surf surf rf F is the magnetic field intensity fieldintensity at magnetic the is is relative permeabil is magnetic compared to its initial value [1]. compared to its initial (3.37) Handbook of Induction Heating ofInduction Handbook of the carbon carbon of the δ in hot hot in ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 tions in differential form can be written as [58–65] as written be can form differential in tions fields,Maxwell’s Maxwell’s electromagnetic Forequa equations. general time-varying physics. applied knowledge reader first-time bylimited the readeror with andof advancedmathematics skipped be can vector and and analysis, calculus, integral calculus, differential including NOTE: 3.4.1 4.1. Section reviewed are in aspects IH. Metallurgical during occur that processes thermal fieldand electromagnetic of concentrate the modeling on mathematical materials composition.chemical as as andFollowing temperature well fieldintensity on magnetic physical depend strongly the of because materials interrelated properties tightly are which analysis, phenomena circuit and heat of transfer,action electromagnetic, metallurgical the desired heating results. heating desired the to improve recipe process induction system ofguarantee and and the system effectiveness and what conditions the heating be design in must accomplished final and transitional the or phenomenon. process model technological properly that the describes appropriate an to choose simulation theoretical is any mathematical step in important the and most Therefore, equations. derived first governing the are the that from results of calculations. accuracy equations) to provideing fail might required that the governchosen mathematical model incorrect (including an lead can to assumptions fied or oversimpli features Underestimation process others. of the in results give unrealistic applications work model and One can certain distribution. well temperature in or initial conditions, material properties, poorlyboundary as such defined sometimes parameters model to chosen of the sensitivity and shouldpossible and consider errors correctness of applied models, aware be and mathematical should limitations of assumptions, the also physics and nature of of process. Engineers the that to have understanding asound essary plexity of the engineering problem, accuracy, cost. and limitations, required plexity engineering time of the model on several depends com factors, the theoretical including choice ofThe aparticular several of hours computational require work can supercomputers. modern which using to formula avery complicated asimple from analysis, hand-calculated vary numerical equation(s). governing the to select simulate and required is that models may Theoretical reduces and curve, development time. new learning systems,factor the avoids when shortens designing surprises, unpleasant most appropriate the establish Computer recipes. process provides modeling acomfort what must helpsIt accomplished be determine also to improve to effectiveness process the heated conditions of component.thermal final and transitional the factors could affect experimentally. nonlinear and of Simulation enables interrelated how prediction different impossible or resolveto difficult could and costly, be which details, cess consuming, time pro error. and determine to quickly Computer trial via simulation enables specialists IH Background Theoretical The technique of calculating electromagnetic field depends on the ability to solve to fieldthe ability on depends electromagnetic of calculating technique The As mentionedAs above, amultiphysical is IH phenomenon acomplex comprising inter factors how computer mayNumerical influence different allows predicting modeling at produce best any can computational that to remember It analysis only important is Before an engineer starts to provide simulation of any starts process, amathematical it nec is engineer Before an anyin set is computational relevantto of a define step quantities first modeling The Mathematical Modeling of the Electromagnetic Field Electromagnetic of the Modeling Mathematical

This section requires from the reader knowledge of certain aspects of mathematics reader aspects the knowledge from of certain requires section This 103 ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 magnetic flux density,magnetic density, conduction Jis current ρ and where

contrast, an integral form is usually applied with the boundary element method. applied boundary the usually with is form contrast, integral an In equations. theMaxwell form of differential a use typically methods difference finite apply ofmethods Maxwell’s forms different For equations. example, elementand finite well by as form applying Stokes’ [58–65,103,104]. theorem calculation numerical Different 3.41) 3.38(Equations but integral form through in differential not in only written be can where i 104

operations: mathematical following the represent symbols system, coordinate these For example, rectangular the in out details. operation the without to carry having differential expression of particular an to shorten vector used algebra in are and popular of ∇ are symbol third the bols along with

The fundamental laws governing the general time-varying electromagnetic field electromagnetic laws general time-varying the governing fundamental The Equations 3.38 through 3.41 3.38Equations through ∇ symbols: of special consist is electric fieldintensity, electric E is , j are the unit vectors X- for unit the the kare , and ∇× UU == = curl i    ∇× ∂ ∇× ∂ U ∇⋅ Y ∇= ∇⋅ ZY HJ UU ∇⋅ E D − UU =+ =− = B ∂ is electric flux density, electric D is ∂ == U gr UU ρ ∂ Z = ∂ XY ij ch di ∂ ∂ ad XY ∂ 0( B ar ∂ t    v D t ge + j ∂ =    ∂ () fr ∂ i fr ∂ () ∂ () U om ∂ ∂ ∂ fr fr om U Z , Y- X U X XZ om om x U ∂ Ga − + ∂ k + Z , and Z- , and Fa Z Am ∂ Ga j ∂ ∂ us ra ∂ U ∂ ∂ U X Y U Y us = da sl y pere ’    + + sl ys aw + ’ axes, correspondingly. ’ k ∂ is magnetic fieldintensity, magnetic H is ∂ k ’ aw · and ∇ · and U ∂ sl ∂ Z ) la    U Z Handbook of Induction Heating ofInduction Handbook charge z (3.40) aw ∂ w ∂ (3.43) , U (3.42) X (3.41) YX (3.39) is electric charge density. charge electric is (3.38) ×. These two useful sym ×. useful two These − ∂ ∂ U Y    ,

(3.44) is B is - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 has two sources: two conductivehas ( physical well. as For interpretation example, Equation 3.38 of curl Halways states the that Background Theoretical where the parameters ε where parameters the medium. isotropic hold additional for and are alinear true The relations constitutive following specified. are field the quantities between relations the become definite equations when These of number unknowns. the than less is equations geometries. of IH workpieces the with irregular with dealing avoid of and in a simple such allow one to explain statementstanding mistakes many will words, or other always end;originate in acontinuous loop. Blines form under Aclear have atthat they points no Blines equivalent source which is zerosaying is to flux density short notation of Equation 3.40 significance. real divergence has thesay To that magnetic of here. packages,” The true small come in particularly is things best “The saying, popular coil. induction located the near conductivefasteners, structures electrically or other induced fieldandfields. aproductmagnetic is magnetic coil induction source of the field the magnetic field of The thetotal coil. magnetic of main of the direction to the tions workpiece the induced fields, in have produce magnetic own which their oppositedirec Equation 3.39. in sign minus by the According to Equation 3.38, eddy currents alternating determined is however, This current; coil opposite is current. coil of that the direction their coil. According to Equation 3.39, source the as have frequency same induced the currents that are that located near the objects in fieldinducesin other workpieceand eddy currents geometry, magnetic coil-to-workpiece the coupling. and changing The electromagnetic current). field’sthe coil current, magnetic That of magnitude coil the on depends strength current source the (coil as frequency have same the fieldwill that magnetic (changing) ing alternat an produce its surroundings in will to Equation current coil 3.38, alternating an According ACvoltage of appearance circuit. coil an the the in in results coil induction to the place application IH. The in of alternating taking processes of some electromagnetic of the applied be space. can result to in any region fundamental field.This of induced that electric direction the Equation 3.39 in sign determines minus take place. The words,other changes area thewhere such fieldin it produces electric an and,in area the surrounding in alwaysinduces Efieldandcurrents produces curling the Equation 3.39, B flux density magnetic conclude rate in one can of a time change that From objects. in surrounding flowing produced currents is whenever electric are there trical resistivity. μ constant trical The 1/ conductivity σ= material; of permeability, the magnetic electrical and Maxwell’s have they not have aconcrete equations only meaning, apurely mathematical The above-described Maxwell’s equations are in indefinite form because the number the of number Maxwell’sbecause form above-described indefinite The in are equations should analyst payAn attention tosimple such Equation as 3.40 relations or 3.41. The undesirablecabinets, of tools, heating an be Equation fixture, 3.38 can there suggests that Let’s how review to 3.38 support Equations basic 3.39 the explanation and used be can , μ r , and σdenote,, and respectively, relative the permittivity, relative J ) and displacement) and ρ 0 =4 JE =σ × 10 π × BH DE −7 =µ H/m [or Wb/(A ×m)] permeability the called is =ε () Oh r µ ε 0 0 ms ’ (3.45) (3.46) charge la w charge currents. A magnetic field Amagnetic currents. charge , (3.47) ρ , where elec ρis 105 - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 Aas vector potential of amagnetic terms in expressed be can earlier. were and discussed meanings have and parameters very similar nondimensional ε are Equation Equation 3.48 as and 3.48 neglected be rewritten be can can than the displacement current density ∂ density displacement the current than 10 much greater Jis induced density conduction the than MHz, less is current currents spaceof (the free vacuum), ε constant the similarly and 106 that

permittivity of free space. Both relative magnetic permeability μ space. of relative permeability free Both magnetic permittivity satisfies a zero divergencezero a Bsatisfies flux density condition magnetic (Equation the Since 3.40), it By taking Equations 3.45 Equations 3.47By and taking account, into as Equation rewritten 3.38 be can For most practical applications of the IH of metallic materials, where the frequency of For where frequency materials, most applications the practical of IH of metallic the And then, from Equations 3.39 Equations from 3.52, then, And and it follows that some 3.38, vector Equations algebraAfter using and 3.39, 3.46, and it possible is to show Therefore, after integration, one can obtain integration,Therefore, one can after ∇× ∇× ∇×       ∇× σ µ 1 E 1 HE r ∇× ∇× =− BA ∇× E =+ D =∇ =− σ / H HE ∂ E ∂ ∂ A    t t    ∇× = , so the last term on the right-hand on the side term last of the , so × =− −∇ =− σ ∂ () . ∂ εε µµ ∂ . σµ ∂ ϕ (3.52) A 0 t (3.49) r t , E (3.54) . 0 0 (3.53) ∂ ∂ ∂ . ∂ E t H (3.48) t 0 =8.854 ×10 . (3.51) (3.50) Handbook of Induction Heating ofInduction Handbook r and relative permittivity relative and permittivity −12 F/m the called is Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 Equation potential. as 3.47 written be scalar can electric where the φis Background Theoretical where 3.50,Equations 3.51, 3.56, and respectively. some derived vector are following equations, by algebra which after the from described be can fieldintroduced. This be field can quency. electromagnetic Thus, a time-harmonic fre asingle with functions oscillating Maxwell’s in harmonically quantities are equations ity. field it possible assumption, is to conclude electromagnetic Therefore, this the that with have asteady-state currents the that model qual by mathematical assuming the simplify of portion heatsources. asignificant contain account can they since into should hysteresis cases, taken losses be these compared In to losses. eddy current quite be significant lacquer can hysteresis and losses coating, before galvanizing, heating valid. is teresis hys the of neglecting Therefore, assumption an nonmagnetic. being temperature—thus cycle, of temperature heating heated surface the the workpiece well above is Curie the of majority applications such the so, is in during because This to losses. eddy current exceed 6%–8% attributed heat compared to not does the typically effect losses teresis extrusion), and to hys attributed aheat rolling, before IH forging, effect and hardening shown saturation, be that itnetic can hysteresis mag the and neglecting continuous and piecewise being as properties material It can be shown that for the great majority of applications, majority IH great forIt shown be that the can it possible is to further relieving, However, stress curing, paint tempering, induction as such some cases, in It should of (such of mentionedmajority IH great forbe steels here that the through as J s =− σ∇φ is the source (excitation) source the coil. induction Taking is the the in density current µµ r 1 µµ 0 r 1 () σ ∇×∇× 1 µ 0 1 J r ∇= ∇= =− ∇= 2 2 HH AJ 2 EE σ AJ ∂ ∂ A j −+ ωµ t j ωσ =− + s ro µ µ J s s o j , ωσ (3.55) σ (3.58) (3.57) A ∂ ∂ , A (3.59) t . (3.56) 107 - - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 system as vector for Equation a2-D potential, of 3.57 magnetic expressed be case the can Cartesian to approximate field 3-D the consideration to a combination of 2-D forms. For example, in A solenoid of the section coil, both cross For example, longitudinal directed. the entirely in fieldintensity, tor electric potential, field intensity)andbe magnetic to assumed be may following factors: to the includes but not is limited (2-D)dimensional Several field factors 3-D discourage assumptions. consideration.This of acombination two- with tend to effectively IH be handled in projects of engineering on. so and impedance, power, current, system induction as of such the parameters design coil required of the accuracy. modeling the would assumption affect not appropriate be when dramatically this can and accurate approximation of applications, majority IH the for some cases modeling are there with the appropriate to H solved be the conditionwith respect can with boundary (Equations equations fields. Thethrough 3.57 field3.59)governing time-harmonic the for and induced and currents impressed the both absent in are harmonics that means quency

where 108 NOTE: coordinates, Cartesian nates. In transverse section, H section, transverse and E and For many IH applications, the quantities of the magnetic fieldFor applications, IH magnetic many of the magnetic vec as quantities (such the considerable problem is 3-D Although there majority agreat in solving, interest practical 3.57Equations 3.59all allow valid forfind and fields through one general 3-D are to In other words, an assumption of harmonically oscillating currents with a single fre asingle with currents oscillating words, other of harmonically In assumption an case), (axis-symmetric coordinates cylindrical In 2. 3. 1. time-consuming challenges dealing with 3-D images and numerical data. numerical and images 3-D with dealing challenges time-consuming input data geometric could and Representation create results well-known of both 3-Dsoftware. must with have user experience The working specific IH). phenomena heat in and transfer properties] [including material of electromagnetic features interrelated tightly account into (especially taking for cases muchComputing 3-D higher are costs Though the time-harmonic representation of electromagnetic fieldis a sufficiently ofrepresentation electromagnetic time-harmonic the Though ∇ 2 vectors have one only component, Z entirely is which is the Laplacian, which has different forms in Cartesian and cylindrical coordi cylindrical and Cartesian in forms different Laplacian, has the which is and B and vectors have one only component 3.55). (Figure allows one This µµ r 1 ∇= ∇= 0 2 A    2 A ∂ ∂ XY 2 AA RR 2 1 ∂ ∂ + XY ∂ 2 ∂ AA ∂ ∂ 2 2    + 2 R ∂ ∂    ∂ 2 ∂ =− AA R 2    + JA + s ∂ ∂ + Z 2 ∂ ∂ A Z 2 j 2 ωσ 2 . (3.60) . (3.61) Handbook of Induction Heating ofInduction Handbook , (3.62) -directed. In the case of a case the In -directed. , E , or A . - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 unit time in a unit volume heat aunit generation). (so-called in obtained time heat is density unit source This per induced eddy with heat associated currents density the source Qis and conductivity, where E FIGURE 3.55 system as cylindrical axis-symmetric forand an Background Theoretical boundary compared to its value elsewhere in the region (Neumann condition ∂ compared to (Neumann its region value the boundary elsewhere in condition) along the or its negligibly (Dirichlet gradient is small boundary alongzero the the right choice ofthe fielddifferent applications.for different representation,could be which E or to 3.62 Equations to E formulated 3.63 respect be and with similar equations can that heat induction systems. of treatment majority conventionalgreat cylindrical a in processes electrothermal the describe fully conditions boundary and initial their with 3.4.2 Section Equation and in 3.63 discussed is equation that Therefore, heat the transfer Fourier by equation: the described be workpiece (time-dependent) ametallic general, transient can the In in process heat transfer 3.4.2 3.55. Figure shown in are applications coils in applying multiturn processes cylinder H to B more be formulated respect conveniently by equations can with governing described the of workpiecesection cross However,transverse in a fielddistribution electromagnetic the theIH system. of section cross fieldin a longitudinal electromagnetic the fornient describing and H , Aand , [1,77,85,103]. electromagnetic for describing used typically Field are that representations is Ais vector potential magnetic the that such selected is region of the boundary The Partial differential equations that are formulated with respect to A formulated respect with are that equations differential Partial vector governing it analogous algebra possibleBy is manipulations, to using obtain B H , or H . Therefore, any given electromagnetic problem may. Therefore, IH of worked A be any in given terms electromagnetic in Mathematical Modeling of the Thermal Processes Thermal of the Modeling Mathematical is the thermal heat, thermal specific the the k is cis metal, of the density temperature, the T is γis field representations in a cylindrical system. cylindrical in a representations , Bfield . Part of the art of mathematical modeling of electromagnetic fields fromderives of electromagnetic modeling of mathematical . art Part of the Cu Cu H + rr rr en en t t “E” or“A symmetr Axis of µµ r 11 0 ” fieldrepresentation y    Bille ∂ ∂ R 2 R AA t 2 adiu Induction coil + c γ s RR ∂ ∂ T ∂ ∂ t +∇ + ∂ ∂ ⋅− () ZR 2 AA 22 kT Z ∇= − intensity mag Dir ec netic fiel    tion of Q =− “H” or“B , H , (3.64) JA d s Induction coil + ” fieldrepresentation j ωσ + Bille .

or E or t are very conve are A/ ∂n (3.63) =0). 109 , or or B - , , Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 plate) as treatment applications. treatment heat and IH in processes heat of the modeling for transfer equations mathematical popular of symmetry formulated be axis along the can nonlinear. highly are surface Equation surface. one from may 3.65, As see boundary heat at workpiece the losses the piece cold contact with or water-cooled rolls guides, etc.), to the normal ndenotes the and heat coefficient, loss under consideration,under α where as expressed be condition can radiation 3.50). boundary (Figure case, the to convection thermal and this In or continuous casting). quenching, partial (i.e., process previous residualof technological the the heat after accumulated preheating, because nonuniform is distribution temperature initial the some cases, temperature. In ambient to the corresponds and uniform usually is distribution temperature initial The the workpiece profile condition time temperature within temperature to refers the at workpiece. at the and at any point in any time initial distribution The temperature 3-D of temperature. 3.48 3.49, functions and nonlinear c are kand both problem. one conclude might As electromagnetic of the Figures a result as solving from 110 workpiece.insulated of aperfectly case applied be the in also condition can boundary of This symmetry. axis words, zero. other is no In heat is at there exchange the of symmetry axis to the normal Equation 3.64 can be shown in Cartesian coordinates (i.e., coordinates Cartesian Equation slab, shown 3.64 be in in heat can transfer strip, or Equations 3.67Equations 3.65 (Equations conditions 3.68 3.66) and and boundary with most the are If the heated body is geometrically symmetrical, then the Neumann boundary condition boundary Neumann the then symmetrical, heated the body geometrically If is For heat most surface owing applications, IH the losses combine conditions boundary condition, the represents Equation 3.64, initial and conditions suitable with boundary In the case of heating a cylindrical workpiece, as acylindrical of heating Equation case rewritten 3.64 be the In can adirection gradient in temperature condition the that implies boundary Neumann The ∂T/∂n is the temperature gradient in a direction normal to the surface at surface point to the the normal a direction gradient in temperature the is c Q γ s is the convection coefficient, the heat surface is C transfer ∂ is the surface loss (i.e., loss surface the is of a result or workas quenching during ∂ T t c γ − = k ∂ ∂ ∂ ∂ T t ∂ ∂ X T T n =    =− k ∂ ∂ Z T α ∂ ∂ XY T ()    TT k sa    ∂ ∂ + ZR T ∂ ∂ ∂ T    ∂ n + +− =    CT k 1 0. ss ∂ ∂ ∂ () (3.66) YZ T ∂ R    44    + kR TQ ∂ ∂ as ∂ ∂ R T    +    k Handbook of Induction Heating ofInduction Handbook + ∂ ∂ Z T Q , (3.65)    . (3.67) + Q . (3.68) s is the radiation the is t = 0. - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 methods, including physical including methods, experiments. of kinds or other semianalytical, analytical, by using or obtained measured be cannot that to produce simulation and of to the accuracy control the information analysts allow will computational apowerful into them tool that to transform and methods of these features contrary, the study On the methods. numerical it to carefully using should them stimulate always approximate not and absolutely from accurate should not engineers discourage solution numerical the fact that is The used. are propriate methods when those results problems). electromagnetic inap and always Therefore, is of danger there obtaining the provide approximatemethods an solution modeled to the problem (including heat transfer numerical all qualities: to point out important one of it their methods, necessary cal is tion on advancedtion simulation methods. numerical to provide is Thus, goal our them. reader ageneral orienta understand the with modeling numerical in who experience have might analysts science limited material and IH so als materi the simplifying while computation numerical electro-heat modern techniques the of nuances computer in-depth modeling. the Therefore, we brieflywhendiscuss discussing literature,engineer/analyst scientific an experienced even easily confused be cialized can advantages and has been used alone or in combination with others. with combination alone or in used advantages been has and certain has methods of Each these processes. heat and transfer putation of electromagnetic com the in used widely are successfully and methods These ment others. and methods, finite volumes,difference, finite element, edgeele elements, impedance, boundary mutual finite as such applyaccuracy, methods numerical effective highly specialists IH modern geometries). sical sufficiently approximate simple in cases results (e.g., obtaining in only clas useful be can methods These methods. modeling seminumerical and simple analytical and formulas complexityuse of the of applications IH increasing have the restricted and significantly Rapid used. are development when methods such obtained be of computer technology mustdesigner aware be applications, many that, can in inadequate results and erroneous The restrictions. 1970sand inherent of the because analyst modern no the longer satisfy 1960s the in popular methods design coil equivalent and circuit methods analytical The 3.4.3.1 3.4.3 isotropic. be to assumed quantity: tensorial the k becomes (i.e., properties nonisotropic composites,multidimensional etc.). laminations, case, this In Background Theoretical Before we discuss the features and applications and of numeri features some most of the popular the Before we discuss Because of the extraordinarily large amount of information that is available spe is the that in of amount large information extraordinarily of the Because and poor manyrestrictions with simplified computational using techniques than Rather However, workpieces, involving of of IH metallic majority be cases great the kcan in dependent, strong exposing directionally is conductivity of some materials Thermal Numerical Computation of the Process of the Computation Numerical Traditional of Methods Calculation k =       kk kk kk 11 31 21 12 32 22 k k k 13 33 23       111 ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 step) (stencils): following expressions the in has (called aspace values of separation distance its and neighboring terms in expressed be relative and of simplicity application.ality to its gener attributed is tool, its modeling popularity and quite FDM auniversal is series. Taylor’s the on utilizing equation based differential approximation partial ofwise the provides method apoint- This area. approximation node its surrounding value with the in variable (i.e., valuethe unknown of the vector potential) or at magnetic temperature an (Equation equations 3.62,ing 3.63, 3.67, or 3.68) difference bythat afinite couples “stencil” of the lines. the of intersections the defined found at equation mesh points by is governing of the tion the quite simple. is approximate algorithm An solu discretization mesh, the orthogonal piece, flux concentrator,etc.) intonumberfinite ofnodes 3.56). a (Figure theBecause of this method for modeling cylindrical or rectangular bodies. or rectangular cylindrical for modeling method this problems. to apply easy It particularly electromagnetic is and heat both transfer solving extensively used for been FDM has The processes. of different modeling for mathematical [89,103–109] technique the (FDM)earliest numerical method was difference finite The used 3.4.3.2 methods. Here, Internet. book or on the weprovided briefly this somethese review only in of extended list reference an in processes electromagnetic and simulation of heat transfer for used computational study most reader popular the interested techniques can An techniques. modeling mathematical different numerous publications describe that areThere calculations. field availableand transfer heat methods for electromagnetic tool. most appropriate the to select analyst the allow computational will This techniques. ing computer of different equipment IH advantages modelthe the to know limitations and problem specific of the or features and phenomenon.the forofdesigner important is It of tasks. for awide program solving variety for one universal to search than it shown that preferable has is rather methods to software apply and methods different numerical of different use the in problems. IH experience for all Our solving optimum is and cases computationalall fits that method not universal is asingle preferred. There are Fields Inc., others. QuickField, and Inc., Inc., Infolytica Software ANSYS,Integrated COMSOL, Engineering SYSWELD, Vector Corp., Magsoft as such companies Inductoheat as software such Inc., by and specialized inside large companies institutes, anduniversitiesresearch Work field in is done this in standpoint. apractical from technology modern of therequirements satisfy solutionsmate,will that but sufficiently engineering accurate 112 By Taylor’s for variables, two theorem value the of avariable at mesh anode can on the The computation procedure consists of replacing each partial derivative computationThe partial of govern each of replacing the procedure consists The orthogonal mesh discretizes the area of modeling (i.e., of modeling area the coil, induction work mesh discretizes orthogonal The of the list exhaustive not does chapter an review of this spaceBecause limitations, application on the depends right choice ofThe computational software and method or software methods numerical problems,For problem each certain of similar or family development. under or are exist programs and methods modeling Many mathematical It is wise to remember that the correct use of numerical methods will provide approxi will methods of numerical use correct the that to remember It wise is Finite Method Difference Handbook of Induction Heating ofInduction Handbook ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3

unknown variables (i.e.,unknown T to solved be respect can algebraic that with of equations simultaneous aset obtain can conditions, one boundary and account into proper the initial taking approximations and local approximation. all local gives the equations By assembling differential fer partial than one on the order one on the of h than simplification in the computational procedure can be used. the computationalbe in simplificationcan procedure neighborhood diagonal), ofthe sparsely the occupied (nonzeros in are only matrices some inversion the Since methods. either matrix by iterative or byobtained direct techniques Rectangular mesh network (grid) used in finite difference approximation. difference finite in used (grid) network mesh Rectangular FIGURE 3.56 mation is on the order on the mation of is h tion error on the order on the error oftion h Background Theoretical

Substitution of the finite difference stencils into the electromagnetic and trans heat intothe electromagnetic stencils difference finite Substitution of the Here, notation the O (i –1,j) Wo hi radiu hj ∂ rkpie ∂ ( ∂ ∂ ∂ ∂ +1 h (i, j) X T (i, j–1) (i, j+1) X X T T ) is used to show that the truncation error involved error approxi to the show truncation the used in that ) is s ⇒ . ce R hj ⇒ ⇒ hi , . 0 +1 2 A O , which is much smaller, is , which to amore leading accurate solution TT TT TT (i +1,j) , ( ii ∂ h ∂ ii ii E − +− + X ) is a linear function. Similarly, function. O alinear ) is 2 11 1 h , T h 2 − − 2 H h − ⇒ 1 , or B + + TT Oh A + Oh ii +− ppro () Leng 11 Oh () ) at node solution each mesh. of be The the can −+ () B ximation 2 oundar h th ofworkpie B 2 oundar () () Ba Real Forw T y () i Ce ck y wa + nt ar ce Oh ra rd dd () ld 2 di i i ffe ffe . ffe

re re re nc nc nc ( h e e e 2 ) is for the approxima for) is the (3.71) (3.69) (3.70) Z (3.72) 113 - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 as Equation rewritten 3.64 be can Z the 3.73 to respect with 3.74 and stencil difference mesh nodes. finite at The vary different Therefore, coefficients of Equations constants. the considered to piecewise are be erties as rewritten (Fourier equation). heating billet equation (Equationdrical governing 3.64) The be can 114 negative effect on the accuracy of the calculation. ofnegative the accuracy on the effect have might computations noncoincidence shown that anoticeable boundaries has of the IH FDM in using appropriate of the in system. IH of boundaries Experience the the regions format: operatorsdifference finite the introducing equation (Equation 3.73) differential by form amore concise in maypartial expressed be

-coordinate can be written as written be can -coordinate For describing the heat transfer process in rectangular bodies (i.e., bodies slab, rectangular in process heat plate, the transfer For describing bloom), and Figure 3.56 shows the rectangular mesh network. mentioned As above, 3.56 propFigure shows material the rectangular the As an example, for cylin an As processes FDM heat for we using modeling transfer illustrate In FDM, it is important that the boundaries of the mesh region properly mesh region of the boundaries coincide the FDM, with that In it important is Substitution of 3.75 Equations 3.76 and difference Equation into 3.73 finite the in results are functions of the temperature. The temperature. of The the statedAs earlier, functions k are factors, among other cand cZ 12 γγ ∂ ∂    k ∂ ∂ Z T    ⇒= c γ c Λ γ ∂ ∂ Z T ∂ tZ T ∂ T tX τ = = ∂ ∂ ∂ T ∂ t cT ∂ cR 11 () ∂ γ =+    cZ 1 k γ    ΛΛ k ∂ ∂ () ZR ∂ ∂ Th ZR T ∂ TT ∂ ∂ ∂ R XY T    ()       +    k ii kR + + ∂ ∂ 1 Z T h ∂ ∂ ∂ ∂ ∂ R T + ∂    R + 1 ⇒Λ       c       1 k ⇒Λ γ kR k ∂ ∂ QZ ij + Y T Z 1 ∂ ∂ (, T , R R T    T TTT (3.75) +    ij R . ττ + + QX 1 (3.76) ). h ,, QZ (, i Handbook of Induction Heating ofInduction Handbook (3.77) + − 1 (, Y ij R ). − ). (3.74) k (3.73) ij , TT ij ττ ,, − h i ij − 1    . (3.78) - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 formulation difference leadsexplicit to finite the to time respect approximation astep-by-step with difference Aforward in manner. directly obtained is step. time each to [106]. implicitforms forms of algebraic aset at solving equations Implicit require forms the of application.on several specifics factors, the including procedure depends numerical advantages disadvantages. choice and of The a particular dependency. time its and own has algorithm Each of nonlinearity features to address these overall calculations. of on the the accuracy effect have concentration can adetrimental current of high areas Rough approximation these in 3.1, Section effect. in skin the including phenomena discussed electromagnetic various governing equation. at conditions accurately least as conditions,the to boundary treat as it very important is equation derived boundary approximation error the approximating the and from erning right). 3.56, (Figure values havethe bottom boundary they nodes real of the at neighboring the nodes are at boundary temperature to the the values the system, corresponding tion then be similar to 3.78 Equations 3.79 and similar be [106]. respectively. Background Theoretical where values. or previously known calculated initially explicitly their from obtained are temperatures to ambient) be appropriateassumed the conditions. and Therefore, unknown the boundary h τ attemperatures time properties, heat material sources, and known of the functions as 1) obtained step are time τ , which is found by the given initial condition (the initial temperature condition is often condition often temperature condition is (the found given by initial is initial the , which The explicit approximation is the simplest technique where the temperature distribution distribution where temperature explicit the approximationThe simplest the is technique problem transient explicit range from heat for formats the transfer difference Finite (transient) time-dependent several available formats are process. There a nonlinear is IH of a“good” importance the factor emphasizes that Another approximation related to is Since the accuracy of the numerical computation depends on both the errors in the gov the computation in numerical of errors the accuracy the the on both depends Since When mesh boundaries do not coincide with boundaries of the components of do the induc of not the boundaries coincide mesh boundaries with When thewill radius to operatorsrespect with differential for expressions difference finite The Approximations of the boundary conditions by Z=ZZ conditions Z=0and Approximations boundary of the As one can see from Equation 3.80, the unknown temperatures corresponding to the ( to the corresponding temperatures Equation from 3.80, see one can As unknown the i , j , and , and are indexes corresponding to the to the indexes corresponding τ are (Figure 3.57a). (Figure step time first the after calculated are temperatures The ZZ =⇒ TT ij ττ ,, + Zk 1 Zk h − τ =⇒ , 0 ij , − =+ ZZ ΛΛ 1 , , Zi j j TT T TT ττ Nj 10 ,, τ ττ jR ,, , jj h − i − + h 1 N T Nj τ − ij 1, = c P 1 = Z γ z = -axis, the the -axis, Q P 0 zN ij τ = , . N (3.80) , (3.79) are R -axis, and the time, time, the and -axis, 115 τ + - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 this parameter varies between 0 and 1. 0and between For varies parameter ξ=0.5, well-known Crank–Nicolsonthis the format not change its magnitude with small variations of h variations small with its magnitude not change [106] as written be format can equations. algebraic of system of a calculation the reduce of any implicit requires computational use method The efforts. 3.57b).choice (Figure of mesh parameters Several were implicit developed methods to more to stable have and compared to results explicit a relatively algorithms independent algorithms. of these properties) use the material may limit nonlinear tially consideration into accurate stable and an essen solution taking (particularly for obtaining versa. vice and approximation used be can difference help. or backward difference aforward For stencil, example, difference of a central instead may stencils of different use the cases, such systems. In or of ill-conditioned unstable case not improve spacesteps and steps solution will the but atypical rather is worsen it. This solution occur. can oscillatory aphysically unrealistic steps. Otherwise, time small extremely leads to condition contradict one usually another. can stability relations The those Sometimes, space intervals, steps, values and time properties. of the material between relations for certain equation or physical phenomena. Unfortunately, accurate stable and are explicit only methods differential independent manycases, the in is, partial and governing of format difference finite time steps h time formulation) stabledifference saidif be is to numerically at small sufficiently finite called approximation numerical solution exact tothe the [106]. format difference finite (also The of closeness the of is a measure format. difference Accuracy explicit finite an when using 116 Examples of simplified explicit (a) and implicit (b) forms for one-dimensional approximation. one-dimensional (b) for implicit explicit forms (a)and of simplified Examples FIGURE 3.57 is a balance between accuracy and stability. and accuracy value The between of abalance ξis choice parameter ofThe the When using implicit methods for modeling heat transfer problems, the finite difference problems,difference heat for implicit modeling finite transfer methods the using When to andprovide ability flexibility of their because more are Implicit popular methods convenience and simplicity of aconcern explicit the ofThus, algorithms, the regardless With explicit formats, to have it not time is unusual a situation where decreasing the ofIt the properties condition should on the depends out pointed be stability the here that The ability to provide ability aconcern The astable solution accurate and numerical primarily is (a) τ

and space and steps h Time steps TT ij “k–1” “k+2” “k+1” ττ Node numbersalongZ-coordinate ,, “k” + 1 h − τ “i–1” ij T i–1 k =+ ξξ () ΛΛ “i Zi ”“ T T . Equation 3.80 has a unique solution. Equation 3.80 solution aunique that and has does TT i k i k+1 ττ ,, jR ++ 11 “i+1” T i+1 k ij +− () (b) 1 Time steps “k–1” “k+1” “k+2” τ “k” () and h and Node numbersalongZ-coordinate Λ Z T “i–1” iij ττ ,, . T + Handbook of Induction Heating ofInduction Handbook i–1 k+1 Λ Ri T “i” ji T T + i k i k+1 c 1 γ Q i+1” τ , j T . i+1 k+1

(3.81) - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 problem. heat transfer transient for the solving used commonly mats are implicit accuracy. desired byfordifference the following finite restricted The step is time The used. steps are time large mesh and appear when coarse too could still oscillations τ+1).levels τand of time 1. complete The when ξ= implicit obtained format is two derivatives between approximation intermediate partial (halfway of an the represents Background Theoretical unknown temperature temperature unknown main restriction for choosing a large h alarge for choosing restriction main + 1 is assumed to be made in two stages using intermittent time step 0.5h time intermittent stages two to using made be level in assumed time τ+1is processes: along the Z alongprocesses: the acomplex 3.82Equations as interpreted 3.83 be of and combination heat two can transfer where ζ and, respectively, be Equation 3.83 will some simple after algebraicand operations, as Equation 3.82 rewritten be can algebraic of equations. simultaneous sets two solving fieldof temperature distribution a known from transition the that means This [106] effectively be used can subroutines to solve 3.84 Equations 3.85. and feature, several simplified computational to this its neighborhood. and diagonal Thanks main occupy the only nonzeros that meaning structure, matrix pied, atri-diagonal having

T ij τ+ , a. A Locally One-Dimensional Format (Proposed by A. Samarskii [106]) Format by A. (Proposed Samarskii One-Dimensional a. ALocally stable; to unconditionally said be implicit is The method however, computational certain After substituting the respective finite difference stencils into Equations 3.82and stencils 3.83 difference finite respective the substituting After The set of 3.82 Equations set step h The of 3.83 time to and stable said be sizes is for all b. Format Peaceman–Rachford [106] As one can see, the matrices of the algebraic of 3.84 the Equations 3.85 matrices see, and sparsely occu the one are can As 05 . . In each direction, Fourier equation is approximated Fourier of equation is necessity implicitly the direction, with each In , ψ , and υare coefficients., and -axis and along the R along the and -axis T ij τ+ , 1 TT is made through the intermediate temperature distribution of distribution temperature intermediate the made is through ij ζψ ττ , + ζψ TT TT ii 05 TT 05 ii ij . . ij ττ TT , ττ ,, + ττ − ++ ++ 05 1 − h ττ ,, 10 . 05 ji ++ , h τ − ji . − 11 h 1 τ − τ ij −+ −+ , ij τ ij is avoiding significant truncation errors. errors. Physically,truncation avoiding is significant =+ , . ΛΛ 5 =+ ij ij , Zi =+ Λ TT 05 -axis [106].-axis level time τto from transition The . λ Zi ττ , jR + Ri T 05 T υ . ττ , υ ji + ii ττ ,, TF 05 ji + ii . TF 1 ττ ,, ji + ττ + + 1 + 1 1 05 , 2 2 ji . =− 1 c 1 c ij ,, =− γ γ c Q Q 1 γ j , Q , j j , (3.85) j (3.83) (3.82) ik τ (3.84) (3.86) T ij k , to an to an τ . The . The 117 τ - - . Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 units. All these can lead to a crucial level of round-off errors. Therefore, refining the space level lead of Therefore, to errors. round-off acrucial can refining these All units. algebraic equations; however, of number arithmetic acomputer alimited only with deals cation error. However,tremendous withof a number associated is mesh fine unreasonably converge as the space mesh becomes finer and time steps become converge steps smaller.time and space the finer mesh as becomes putation done is correctly, variables (e.g., values the unknown of the temperature) should as follows:is thethumb com rule if of The insignificant. is calculations between ference dif the until steps time for mesh or smaller afiner calculation to repeat the it necessary is shows comparison then compared. difference, the are alarge If results the steps and time and provided mesh sizes are different for the calculations computational The experiment. of variation variables. insignificant where is mesh forties) there areas coarse and ofdients variables (i.e., fieldintensi vector or magnetic potentials, temperatures, magnetic gra greater Naturally, the with for regions mesh size asmaller to it select recommended is steps, among mesh size, computationcompromise time of modeling. accuracy and time, solution. Therefore, should areasonable there be more complicated time-consuming and a addition, of in number recommended. nodesIn large results are the that it quite clear is for accuracy, greater steps techniques, anyspace numerical of time the in and smaller as the however, FDM. when Certainly, using FDM critical in become particularly parameters these techniques; modeling any numerical of the using calculation of the stability and accuracy computer ashort execution time. memory and aminimum requiring matrix tri-diagonal of the account into features the takes and quite effective is algorithm This method. elimination two-step Gaussian the is matrix for atri-diagonal solving ods most of etc.) widely meth the One methods. overrelaxation used techniques, direct and equations: iterativeferential (such Jacobi the algorithms as method, method, Gauss–Seidel dif intothe partial stencils difference finite of substitution the after obtained equations to 3.84 Equations 3.85. and similar format Peaceman–Rachford is the R implicit Zand in direction explicit in 118 Correlation among the round-off error, truncation error, and mesh size. mesh error, and error, truncation round-off the among Correlation FIGURE 3.58 It is wise to remember that the reduction space of the the that steps to leadsreduction remember of to the trun It wise is by determined usually steps is time and of combination mesh parameters optimal The on the a pronounced effect choice steps has of mesh generation time and optimal The for algebraic solving mentioned used As earlier, have there general techniques two been R explicit Zand in Equation 3.86 direction implicit is in Error TT fine mesh ij ττ ,, + 05 Ve 10 . − ry h τ ij . 5 erro Round-off =+ ΛΛ . A set of algebraic equations that corresponds to of corresponds algebraic. Aset that equations r Zi Total erro TT Mesh size ττ , jR ++ 05 . Tr r uncatio erro ij , 1 n r c 1 γ Q Handbook of Induction Heating ofInduction Handbook Coarse mesh . However, Equation 3.87 is iik τ , (3.87) ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 proposed finite element models are similar in form. It shouldthatbe mentioned here element are similar finite proposed models P. field computationSome theimpossible the contributions. of it makes of all to mention processes. to simulate electromagnetic 3.4.3.2, Section in trated a heat for modeling here we transfer of FEM demonstrate use the illus of consideration been into FDM use has the that phenomena taking and transfer heat phenomenon. and acomplex is IH Since transfer of combination electromagnetic forweighted computation very effectively used been residuals formulation has of heat problems. suited Forfor example,might certain better be the elementfinite techniques provides FEM the element-wise an approximation Certain equations. governing of the [104,110–125]. FEM decades) development the boosted three last has the of of several the variations within improvement tremendous The tasks. computer in capabilities (particularly engineering and of scientific tool for numerical most avariety It popular become the has engineering. Later, engineering. applications of of have FEM mechanical areas to other in expanded applied was originally technique versatile numerical IH. This in encountered those ing problems, approximate an includ solutiondevoted technical for different to obtaining group techniques another modeling of is element (FEM) numerical finite method The 3.4.3.3 butdiagonal five-seven-diagonal. or no longer of tri- be algebraic will approximation. equations local amatrix therefore And of number nodes of involved the increase the an in approach in results this time, same the error, approximations. Such approximations but truncation reducing allow difference at may improved be calculations by numerical of employing the accuracy finite higher-order difference (Equations approximations through andsecond-order first- finite 3.69 3.72). The meshes. moredifferent, it different, often efficientphenomenon-optimized use is to one mesh generation. to save on the However, time physical the if phenomena inherently are approach would and allow atime-saving like seem might mesh. This universal a single use etc.), electromechanical, heatmagnetics, transfer, transformation, it very phase attractive to is niques as well. as niques putation valid not for only are FDM errors but tech of numerical majority for other the metic is a helpful way ahelpful is metic to avoid computation by caused errors. round-off failures become excessive errors round-off the (see 3.58). Figure of adouble use The arith precision improve computations, of steps can the accuracy the time the unless reducing mesh and Background Theoretical The large number of papers on the subject of of applications FEM number papers large subject The on the for electromagnetic 3.4.3.2, Section in FDM provides the described As apoint-wise approximation; however, follows: as are Some of these used the most on commonly illustrated based been has differences of finite method The When modeling processes that couple that processes physical modeling several different When phenomena (e.g., electro The above-mentioned remarks regarding different aspects of mesh generation com and aspects different above-mentionedThe regarding remarks • • • • • Methods based on minimization of the energy functional, and others and functional, energy of the on minimization based Methods Pseudo-variational methods method Galerkin of the types Different method Ritz of the kinds Different Weighted (weak residual method equations) governing of form the Finite Element Method 119 ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 the accuracy of numerical approximation and allows reduction of the required total total approximation of numerical accuracy allows reduction required and ofthe the element). torus elements improves isoparametric of high-order use The triangular called (the section so- cross atriangular has and of revolves symmetry ring Each axis the around elementrings. set mayof case,as a afinite such mesh represented be cylindrical metric shapes allows them, in fact, to satisfy regions of practically any geometry of system. induction of any geometry practically regions fact,shapes allows in to them, satisfy at of every energy node of element. each providestional minimization func (numerousthis subareas elementsof or finite mesh);therefore,minimization the 2-D field eddyconditions. current boundary problem respective with respectively. current, source rents, the and field, magnetic ofeddy the energy cur the represent integrand the inside terms third and ( condition condition) or Neumann (Dirichlet boundary along zero the Ais tor potential vec(Equation 3.62). magnetic the that so selected be can region of the boundary The equation by differential a2-D described is partial Jand density current of the direction [110,111]: following form the in written be region. value the the elsewhere in system. cylindrical axisymmetric (the solution the solver)and techniques were 2-D created for both system) (Cartesian and element ofequations, finite aset to obtain its minimization functional, energy oftion the for example, at vector node. each potential formula magnetic unknown, The to the respect to solve algebraic with of equations global simultaneous set it the necessary assumption, is that result As a of unit length. stored of fieldper the energy minimum to the responds rounding area. workpiece,and includes the sur which conductiveing, fixtures, coil, tooling, electrically of over model area total integral for the the minimized is directly. functional energy The equation (e.g., governing to the Equation 3.62 or 3.63) equation of that solving instead that corresponds functional energy the fieldnetic minimizing computationby is obtained S. Gurevich and coworkers and S. Gurevich [103]. element concept by proposed W. Lord [113,114], S. Udpa coworkers and [120,123], and finite- the approach on acombination of based one of is form description FEM. This J. W. Sabonnadiere, Trowbridge, ashort following is The others. many and J. Simkin of was FEM provided by W. Lord, A. Konrad, S. Salon, S. Udpa, J. Brauer, A. Bossavit, inputFEM. intothe earlier Markedusing development fieldanalysis lation of magnetic [110,111]Silvester M. and Chari formu variational general nonlinear first the presented 120 where ∂ A The simplestThe 3.59a). 2-D first-order(Figure triangle the elementis finite axisym the In their of possible.are elements shapes of and finite Flexibility arrangements Many geometric According of to study FEM, divided area nonoverlapping into is the of number finite (Equation 3.88) solution to the functional of corresponds the of the minimization The The energy functional corresponding to the 2-D to the equation corresponding (Equation governing 3.62) functional energy The can A vector potential Magnetic The principle of minimum energy requires that the vector potential distribution cor distribution vector the potential that requires energy principle ofThe minimum Because of the general postulate of the variational principle, variational solution of the the general postulate of of the Because electromag / ∂ = 0), meaning that its gradient is negligibly small along the boundary compared to boundary alongn = 0), the its negligibly that gradient is small meaning is the total area of modeling and J and of modeling area total the V is F = ∫ v    22 µµ 1 r in the 2-D case (longitudinal cross section) acts in the the in section) acts cross 2-D the (longitudinal in case 0    ∂ ∂ X A 22 + ∂ ∂ Y A S is a source current density. current asource first,second, is The    +− jA ωσ || 2 Handbook of Induction Heating ofInduction Handbook JA S    d V , (3.88) ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 geometry (if applicable). (if geometry element from element to element. each different but be can within to constant be postulated are Material mesh. properties a finer require regions of presence ferromagnetic and frequency (i.e., where rate Higher unknown ofvector the the high. potential) of is magnetic change the element to relatively be mesh has finite fine elements the regions smaller)mustof be in (sizes ment. However, solution, numerical of reasonable the finite accuracy order the in to obtain of the mesh. of the density interaction. Noticeend to plate reduce anonuniform coils their located is between Copper bars. of long heating titanium for in-line used system IH is elements. This gular trian solenoidshows elementusing exampletwo finite an ofmesh of the coils multiturn FDM. with similarities (mesh generation) elementdiscretization some finite thathas regarding general remarks problems. effect of skin treatment problems allowbetter they because and electromagnetic thermal both in modeling used elements finite popular increasingly complexity. Six-node eight-node and 3.59b) elements (Figure isoparametric have become computation algorithm and in time of some increase ofnumber elements at expense the Background Theoretical Finite element mesh for two in-line multiturn coils with copper end plate located between coils. between plate located end copper with coils multiturn in-line two for mesh element Finite FIGURE 3.60 (b). elements (a) isoparametric eight-node and six-node versus triangle First-order FIGURE 3.59 It is beneficial to take advantage of the symmetry and/or advantagetake It to beneficial is the symmetry of the modeledof periodicity The size, shape, and positioning of these elements frequently depend on personal judg elements depend frequently on size,personal of shape,The these positioning and of study subdivided area nonoverlapping is into The elements finite (mesh). 3.60Figure some following are The of analysis. FEM aspect avery important is Space discretization (a) (x,y) A(x,y) –α A n n 1 +α l A l 2 x *x+α y m 3 *y A m 1 (b ) 2 Si elemen x-no 6 3 de Isoparametric elemen t 4 5 3 1 2 ts Eight 4 elemen 8 -no de t 6 6 5 7 121 - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 values of A vertex by equations assuming independent simultaneous three the from calculated be can as defined be can atriangular within distribution vector potential magnetic elements, the then triangular order parametric first- are elements finite chosen element law each the supposing that across and a linear 3.62 (Equations vector potential 3.63). and to magnetic respect equation with governing the solving in tion), used been has FEM the sec cross workpiececylindrical (longitudinal of a profiletemperature alongthe length 122

as element, written be can any triangular within functional energy of the mization values of A unknown to the respect with equations algebraic elements. aresult, simultaneous of of As all the aset energies of the sum the equals area modeling entire an with (global) associated total The energy minimization. it convenient is to apply functional, of the element-by-elementby-node minimization node- the node, to each to zero. of respect performing equal Instead with functional the derivative partial first of the by setting arranged be at can every node. This functional area. modeling the approximation of interest, an area for it possible total Athroughout is the to obtain elements element. represent approximation that the the alocal inside By to all extending at every element, node each potential in value the it of possible Aat is to obtain any point vector magnetic of elements the and geometry the Knowing area. triangular the twice as rewritten be can of equations set Therefore, local the Assuming that the local behavior of the electromagnetic field can be field approximatedcan by behavior electromagnetic of local the the that Assuming and fielddistribution electromagnetic the to obtain when cases, many it necessary In is Based on 2-D linear approximation on 2-DBased laws, linear α coefficients the The matrix notation of this set (Equation set 3.90) notation as of this matrix written be The can Energy balance within the modeling area is determined by minimizing the energy energy the by minimizing determined is area modeling the within balance Energy Equation 3.91 in avalue matrix as introduced be square of of the can determinant The After some algebraic operations, the local matrix equation, which represents the mini the equation, represents which some algebraic matrix operations,After local the l , A m , A n at the three nodes of a triangular. nodes of atriangular. Aat vector three the potential of amagnetic       A A A AX m [[ n l AX (, AX AX Vj mm nn       ll ][ = =+ =+ =+ ee YX +=       αα αα αα ). =+ 12 12 12 1 1 1 WA αα XY 12 XY XY ]] mm nn ll [] + + + α α α + []       3 Q 3 at each node can be obtained. at be node each can 3 Y α       Y Y l n 3 m e α α α Y . , 2 3 1 (3.92) (3.89) (3.90)       . Handbook of Induction Heating ofInduction Handbook (3.91) 1 , α 2 , and α , and 3 are constant and and constant are - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3

where Background Theoretical as written be equation can conditions, aglobal matrix boundary triangular. of aparticular area across-sectional Δ is problem (i.e., system) are cylindrical 3.98through [114]. of Equation 3.92 matrix Parameters local of the foraxisymmetric the andto easy apply.approachis effective very artificial new Such an diagonal. the times conditions replaced valuesright-hand are by sides of equations known boundary of those is known by a significantly large number large number by asignificantly (e.g., known is 10 nodes vector where value the potential the magnetic of the representing equations of the terms multiplying diagonal the requires computational technique This diagonal. the ing blast Equation called 3.98. is in conditions boundary techniques most of popular the One

After assembling all local matrices of finite elements and specifying the corresponding the corresponding and elements specifying of finite matrices local all assembling After For the axisymmetric case, the local and global matrices will be similar to3.92 Equations similar be will global matrices and local case, the For axisymmetric the It is necessary to mention here that there are several commonly used ways to specify the the ways several used commonly are to to specify mention there here that It necessary is       bb cc aa [] V lm lm lm e = b c a 4 n n n µµ       r 1 = 0       ∆ ()       XY ( () () mn [] () b () XX bb W bb YY mml nl ll nm bc ml − − [] e −− [] ++ + ++ Q [] = A XY GA cc cc e ll ml nl ωσ nm e [] 12 cb = = )( ∆ J       S 3 ( =      () ∆ ( bb A A bb A XY mm n [] 21 11 12 lm      ( m Q n bc l () nl XXX YY mmn 1 1 1 30       ln nl . ++ . ). At the same time, the corresponding ). corresponding the time, At same the      −− (3.98) − −−

cc cc (3.96) 2 1 1 XY lm mm cb ln      mn )( (3.95) )( )( )( () bb bb XY mn ln bc lm () XX nn YY + + lm ml − cc cc ln mn XY c n ml ) ) )       ) (3.93)       (3.94) (3.97) 123 - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3

124 as conductors determined be in can field. output electromagnetic For of example, the parameters induceddensity thecurrent therequired region, itof possibleall modeling is the to find in vector potential magnetic solution, appropriateanalytical physical models, of or experiments. results available the approximation FE with of on its the accuracy comparison based obtained the application.of an stage the of code, developing FE In adeveloper testing and judge can 3.58). shape specifics depend and on the size of number elements their and optimum The limit might improves be a equations Figure mesh therebut (see afiner with differential tial into considerationinto banded symmetry. and matrix’s the nature sparse taking while inversionsolved either techniques iterative matrix using or direct methods Equation 3.92 in Aglobal equations. matrix be system can large ofof matrix the solving where R After solving the system of algebraic equations and obtaining the distributions of the of the distributions the system of the obtaining algebraic solving and equations After As mentionedAs above, approximation numerical par of governing of the accuracy the the the computationof portion FDM, significant the with the As work forFEM consists c is the radius of the finite element radiusand finite centroid of β the the is       [] V bb cc aa lm lm lm e = 4 b c a µµ n n n R r       c = 0 ∆             () RZ ( () βββ () ββ ββ mn [] () () ml W nl RR ZZ ll nm mn [] e Q − + ++ ++ − −− = [] cc cc cc RZ Jj A e ll nl ml e ωσ nm = =− e 12 )( = JR R Sc ( c       ββ () 3 β ∆ ωσ ββ ( mm A RZ n A A lm ∆      βββ () ( m nl n l A mn ZZ RRR      21 11 12 .       ln nl ++ 1 1 1 (3.104) −− − −− (3.103) cc cc cc      lm mm RZ 2 1 1 (3.102) ln mn      )( )( )( (3.101) i )( =b ββ () ββ Handbook of Induction Heating ofInduction Handbook RZ mn ln β lm () i +2 ZZ RR nn lm ml + + − Δ cc cc cc ln mn RZ /3 ml n ) ) R ) c       , i=1, m ) ,       (3.99) (3.100) , n - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 [113,114]: follows as nents B nents these, special FEM subroutines have subroutines FEM developed. been special these, moreis convenient voltage the to use input the parameter. as coil of the as such For cases (e.g., some cases in time, same parallel),the in it electrically several inductors connected applications. for At hardening induction case the often input is the parameter. as coil This powertotal on. loss, so and impedance, coil flux density: magnetic vector the of and product density computed cross vector of the the current of from total

Background Theoretical

From Equation 3.106,as be obtained can flux the density The above-described FEM requires using the current density or current of the induction induction of the or current density current the using requires FEM above-described The stored as energy, such possibleIt to compute also is characteristics flux other leakage, the conductors workpiece the and be can current-carrying force in density Electromagnetic as calculated be can fieldintensity Magnetic workpiece, compo flux ofdensity acylindrical magnetic case the For axisymmetric the The magnetic flux density components flux B density magnetic The The total current density in the conductor, the in density current total The Electric field intensity can be calculated as calculated be can fieldintensity Electric R and B and Z can be calculated as calculated be can B FJ RZ xy ∂ ∂ =− =× A Y BB =− JJ ∂ ∂ =+ A Z =− H Ej BF   B ;. =− = ;. s xy xy 22 x ;. and B and µµ B r0 jA B ω B yx ωσ ∂ ∂ A = X A =−   . . 12 ∂ ∂ (3.110) (3.109) / = A y R . JB can be calculated from Equation from 3.52 calculated be can (3.105) B . × (3.107) + A R (3.106) (3.111) (3.108) 125 - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 heater workpiece and 3.61), (Figure subdivided appropriate into are elements. with As Ref.in [127]. solution The of Equation 3.112 its simplest in [129] form here. presented is Q kind: equation second of the integral system induction Fredholm bythe the conductive of domains electrically the in densities current to the respect with described be a systemcan in such field billets).distribution electromagnetic or titanium The aluminum, workpiece nonmagnetic (i.e., axisymmetric placedple. copper, an are around coils Both driven by voltage and asinusoidal series involved are exam source. No the harmonics in Ref. in [129]. approach on giventhe here is based discussed to MIM tion Tozoni [127], P. Dudley and R. Burke [128], Abrief introduc several researchers. and other E. Kolbe W. and 1962 in Reis [126]. was done development by O. Further technique of this was by technique this problem. IH simulation of describing the texts earliest of the One do FDM, formulations and not generally have integral (such space to consider areas air). free as FEM Unlike mesh of discretization. the conductive system IH of limit the bodies electrically (computation)tion words, conductive other In of the to domains. electrically surfaces limited is computer execution time. memory and less requires typically equations, which formulation electromagnetic of the a differential equation approach of applies using integral systems. MIM instead an for for IH cylindrical of amount large computational work. unrealistically with associated is that enormously required an mesh long is large since systems eling conventional complex connections. and circuit or multilayer single oras to multiphase asingle power solenoids with connected source the of be application,fabricated specifics can on the coils 1m. Depending as large as be can of some coils diameter 10 long inside exceeds as 30m. be The as can sometimes mand 2.19). (Figure coils of one consist heaters or several can inline of a system length total The 3.1). shape (Figure or rectangular solenoid of cylindrical multiturn coils Such induction as designed to Induction inductors billet/bar for hardening. surface heaters typically are compared quite different are extrusion and rolling, upsetting, forging, working including inductors hot involvedThe and warm before metal of bars, IH billets, slabs, like in the and 3.4.3.4 126 where represents the heated the heater induction workpiece); Wrepresents the and H represents V and source voltagesource element. of the value The of V tion of the elements (current-carrying rings); elements of S tion (current-carrying the and P elements Qand between inductance mutual ; J The electrically conductive induction the system, induction of the including regions electrically The described most complicated the been equation has in case integral of the solving method The Let’s first consider two axisymmetric multiturn coaxial coils (Figure (Figure 3.61) coils coaxial in multiturn Let’sconnected two axisymmetric consider first devoted of formulations MIM several FDM different FEM, are to the and with there As Another advantage of using integral equations deals with the fact that the area of the integra of the area the fact that the with deals advantage equations Another integral of using used be can (MIM) method alternative to an FDM FEM, impedance mutual As and the Two of mod challenge the well-known disadvantages of FDMare FEM the of and both Q and J and ρ Q Mutual Impedance Method Mutual Impedance = 1/ = P and P elements Qand the in densities current the are σ Q is the resistivity of the element of the Q resistivity the is 2 πρ RJ QQ += jM ω PH ∈ ∫ , W Q , respectively, interac amutual representing is zero for all elements workpiece. of for zero the all is P are computation are ( areas QP ; JS R PP Q d, is the average the is radius element of the V Handbook of Induction Heating ofInduction Handbook Q

, respectively; M p ∈ H , W QP , where Q is the the is is the the is (3.112) - - - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 as rewritten equation solenoids. (Equation 3.112) integral considered multiturn to The act as be can element.each within to constant be assumed are properties material FEM, and densities eddy current Background Theoretical tively; M and sents the induction heater. induction the sents workpiece. Hrepre cylinder of the half top the Wrepresents MIM. for system induction of the Representation FIGURE 3.61 where r the impedance equations for this case is case for equations this impedance the (elements 3.61, Figure shown in sketch to the of elements two system workpiece of IH consists the the obtained. be can equations of system, induction global impedance set the the conductive elements electrically oflaw. to correspond all that equations assembling After well-known Kirchhoff’s the equation representing converted impedance is an into kind coils ( coils given in Refs.given [130–132]. in ( rings of the resistances The elements. current-carrying the of interaction all the representing inductances

(3.114) where If the skin effect in the coil is pronounced, then the multiturn induction coils can be be can coils induction multiturn pronounced, the is coil then the in effect skin the If A set of global equations representing the induction system is obtained below. obtained system induction is the of representing According globalA set equations 3.112 Equations from seen As 3.113, and equation second of the integral Fredholm the The formulas for calculation of the various self-inductances and mutual inductances are are inductances mutual and self-inductances various of the for formulas calculation The jM ωω () rj () QQ jM r ++ ++ n M and r and nQ Q ω = resistance of the element of the Q =resistance QQ ωω and Q P and ++ MI , MI PQ M QP m QQ I mQ ) can be calculated as calculated be ) can PP , Q M Q , R M Q Qn Q connected in series. The global set of global set The series. in n connected mand coils induction two ) and nn , M , and , and l Q Qm Axis ofsymmetr Wo , M M jM rkpie dQ PQ ( mm rrj ( PP , are the self-inductances of the elements coils, of respec self-inductances and the the are M ce jM ++ rI nnP QQ Pn ωω ++ , MI QP M y += MI Ij . Pm jM r PP PQ ω mP Q )( , PH = )( M ∑ ∈ Pn 2 nQ , W πρ P dl , QQ M QQ ++ QP R nP , IV PQ M (3.115) rr nm r Q ++ , and r and M jM , mm ω (3.113) mQ () Induction heater MM , jM M ω P m Pn ) and the resistances of the of the resistances the ) and nQ () mP V += oltage , and M , and MI nn Pm += mm (V MI ) I ) n mn mm mm n are the mutual mutual the are ), n = 127 V 0 0 - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 M where 128 where currents. of the matrices column the workpiece; as n coil of the I components ( imaginary have and real both currents the that showing matrices column (Equation of equations set 3.117) the and symmetric as is rewritten be can

a 3.119) developed been has [129]: computational (Equationof equations set procedure formatrices, the a special solving computer for order and storagein to reduce execution time memory required the of all where a r WH +j Since the matrix S matrix the Since Upon evaluation it inductances, mutual for obvious equations the of becomes that the The set of equation set (EquationThe 3.118) as rewritten be can After some simple algebraic operations, the resulting matrix equation can be rewritten rewritten be equation can some simple matrix After algebraic operations, resulting the PQ , a =M I x HW ); V and K S W are matrices of the mutual inductances. mutual of the matrices are space r QP is a diagonal matrix consisting of the resistivities of elements of coils and the of the elements and of coils resistivities of the consisting matrix adiagonal is , a and a and H S . is the number of turns of number turns the Nis and coil of the winding space the factor is turn of the are matrices of the self-impedances of the workpiece of the self-impedances ofand the coils induction matrices and are x is a square matrix of the self-inductances and mutual inductances; mutual I and self-inductances of the matrix asquare is r , V WH x are column matrices of the voltages of the matrices column are ( =a r is a diagonal matrix and the matrix S matrix the and matrix adiagonal is HW . Therefore, the matrix of . Therefore, matrix the   SS [] rx SS +     rx     aa aa SS SS ++ HW WW xr rx SS jj −− rx r 11 n     − [] = aa II aa HW rx   WW H lK H nn 2 []     δ IV πρ     xr     nn =−     I I sp R I I x r ac W   =+ H     H e H     [] N VV =     = rx ,     (3.116) SS V V     xr V j x r V = 0 H     x .     is a symmetrical square matrix, matrix, square asymmetrical is V (3.119) , , (3.117) Handbook of Induction Heating ofInduction Handbook (3.118) r   r +j (3.120) V x ), which are similar to ), similar are which r , I x are are I = Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 BEM preferable. FDM and is of acombination BEM cases, such or FEM In and of processes. appreciablycases nonlinear interaction. electromagnetic their and reducing fieldcoils between electromagnetic ductive the noticeably end plate coils located distorting is between that con electrically an alloy Ti-6Al-4V is There positioned coils. induction inside is titanium BEM. made using bar from Asolid cylinder obtained coils multiturn of in-line two ings materials) make this technique quite attractive in certain applications. certain quite attractive in technique materials) this make nonferrous with of when mesh generation (particularly dealing accuracy good well as as simplifies meshing. substantially global solution. This (i.e., qualities surface along surfaces) densities unknown would the current that satisfy 3.62). induction system of (Figure the domains A computational the procedure establishes regions. in external fielddistribution the problem describes external The thebody the of workpiece.within field distribution electromagnetic the problem describes problems. iterative Using an tromagnetic solved. be can internal procedure, The tasks both elec internal and may external divided of be tasks: two into processes induction modeling problem case, the of mathematical this of In interest. area of over the gral boundary the BEM has similarity with MIM. with BEM similarity has respect, computation. the conductive in this consideration In domains electrically only into allows one to take This integrals. the values in explicit boundary of appearance the the to is equationcomplete integral The as is, propagatingthanks it regions. infinitely conditions at forassumption boundary artificial problem. an to make It not is required of form Maxwell’sgral equation forelectromagnetic a governing as the used equation is nal articles [133–141] articles nal of modifications various BEM. describe that and jour several find texts,proceedings, conference reader interested will text. The of this beyondFDM, BEM of is scope the its FEM, forms the and or Athorough MIM. discussion for needed BEM those more the are advanced than to discuss required mathematics The late the 1980s in to widely IH for early be to modeling the 1990s. used started method This devoted BEM. computation to IH the techniques of numerical called is family fourth The 3.4.3.5 applications. IH in method of wide this use the particularly, limits in FEM. This element (BEM) method boundary [77]. the with method inductance mutual the power, system induction of including the parameters design efficiency, coil andso on. power well as as important eddy other densities, and heat currents distribution, source Background Theoretical BEM sometimes faces challenges in providing required accuracy of computations accuracy in providing required in faces challenges BEM sometimes As an example,the an surround As field 3.63in Figure magnetic of shows the adistribution Such advantages reduction of the computation as simplicity, time, user-friendliness and conductive electrically of the at boundaries required only With is BEM, meshing the With the BEM, unknowns of the electromagnetic field are expressed in terms of an inte an of in terms field expressed are electromagnetic of the With BEM, the unknowns Here, we just mention when that applying BEM (in contrast to FDM or FEM), inte an provides shows not accurate typically as a solution experience MIM that FDMOur as or, was extended computation for MIM the The workpieces of of IH magnetic by combining of 3.120 Equations set the solving After 3.121, and and currents coil the obtain one can Boundary Element Method Boundary [] IS rr =− [] − 1 [] VS rx I x . (3.121) 129 ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 which are used in solving the thermal problem assuming that the electrical resistivity and and resistivity electrical the that problem assuming thermal the solving in used are which the of Joule distribution heatthe sources, step, first calculating the allowing during obtained are characteristics approach. a two-step Electromagnetic called simplest is The method example, exceed 900 s, easily heaters size. for can on billet billet induction cycle time depending longer. substantially are applications, IH many For processes in time, heatsame the transfer 0.02than s the less frequency).on applied significantly (depending constants the time with At very fast, are processes have IH orders in of Electromagnetic different magnitude. processes heat and coupled transfer (time constants) scales electromagnetic phenomena. of the time The able of are developing that necessity to computational with deal the algorithms tating special following: 3.1.1 (see properties Sections material of the nature interrelated the 3.2.1) and the as such coupled phenomena tightly heat are and of transfer because electromagnetic the Both 3.4.3.6 3.60). Figure (compare with coils multiturn in-line two for mesh element Boundary FIGURE 3.62 130 Magnetic field distribution around two in-line multiturn coils. multiturn in-line two around distribution field Magnetic FIGURE 3.63 There are several ways are phenomena.There heat and to transfer couple electromagnetic the Obviously, dic nonlinear, highly process IH of the physical variations make properties these • • frequency (Figures 3.8 (Figures [a],frequency 3.9, 3.10). and fieldintensity, of magnetic afunction and temperature, is permeability Magnetic 3.5 3.48 3.3 and 3.50). through (Figures perature through of tem functions are conductivity, heat, Specific thermal resistivity and electric Coupling of the Electromagnetic Thermal and Problems Handbook of Induction Heating ofInduction Handbook - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 the heat cycle, meaning that the electromagnetic properties remain unchanged. During During unchanged. remain properties electromagnetic the heat that the cycle, meaning of intervals time certain notduring significant are variations temperature that assumed iteration for 3.64). an (Figure process calls method approach,This According it to this is weakly coupling the method). coupling (also method called indirect the called enon is heating applications dealing with nonmagnetic materials. nonmagnetic with applicationsheating dealing of low-temperaturenumber limited in a used be approachThus,can this errors. significant in result can and avery rough cycle is postulation heating entire the during properties (e.g., materials ferromagnetic with ing steels), carbon material of constant assumption an than eight times. Furthermore, μ Furthermore, eight times. than 3.1.1Sections 3.1.2, more and of process heating the during vary can ρ of some the metals 3.10 approach. shown atwo-step 3.48in As and 3.51 using in danger through the reveals modest and computerits short execution time memory requirements. for approach known two-step heating. is The during do not change permeability magnetic Background Theoretical Block diagram of the indirect coupling process. coupling indirect of the diagram Block FIGURE 3.64 The most approach common phenomThe heat to and coupled transfer electromagnetic Even a cursory look at the behavior of the material properties in Figures 3.3 through 3.3 through Figures in lookEven properties at behavior material of the the acursory Material prop Cu obtaine Te distribution rr obtaine Joule loss mp ent densitie profile her and ar is erature e d her e d er tie e s sS r can vary more than 100 Therefore, when more times. deal than vary can Ele Ele He He convergenc convergenc convergenc ctr ctr condition condition condition at transfer at transfer Result Globa omag omag netism s l neti e e e c iterations iterations Internal Internal yst em ge ometr iterations Ex te rnal y 131 - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 and μ of ρand the of variations (because significant become step. heat conductivity variations the at as source time soon each As thermal and computation to heat update used values transfer is the heat of specific time-stepped the workpiece from the obtained within distribution heat temperature the The sources. ing without continue correct calculations the transfer heatintervals, sufficiently small those 132 development, can be considered useless information. Such information might be of be might interest development, Such information information. considered be useless can nonconductive air. the as such areas, to generate amesh/grid necessary equation, it also is electrically differential within erning gov of the Unfortunately, continuity area. and criteria of conditions to suit smoothness the organization. program of problemtype and the two on depends the methods of efficiency of the comparison one would As a of obtained. global expect, algebraic set is way the equations which in method).energy difference finite (the so-called minimization functional of aform as described be fact, also In FDM can afunctional. duced away as to minimize more precisely.shape boundary field FEM In computation,intro often are finite elements do not overlap one another, do not have nodes outside fitthe and complex boundaries, the geometry, could they have components. nodes of outside the elements Finite boundary the fact, are, in quiteTherefore, similar. methods two the algebraic to values. to its solved be of nodal equations simultaneous aset respect with in (e.g., acontinuous function or vector temperature)discretize potential magnetic result and statement provides avariational and with element-wise approximation. methods Both derivatives equation. governing starts FEM the approximation partial in the replacing provide stencils a point-wisedifference above, outlined finite As that FDM requires advantage over adistinct has FDM. FEM instance, this In and forms. materials of mixture of a thecase in or configurations ary the modeling algorithm is simple is fast. and algorithm modeling the grid, orthogonal of the Because or rectangular. simple cylindrical as has such geometries, application.induction It easy, is for example, to apply area FDM modeling when the the the particular of over specifics on the depends of another one technique selection the that lookEven at network 3.56 acursory (Figure meshes vs. 3.60 Figure vs. 3.62) Figure reveals 3.4.3.7 where memory intensive. it is absolutely cases and is It in only time should used be needed. solved intensive extremely simultaneously. computer an coupling in results execution Direct (i.e., process be will thermal of that eters the of aglobal matrix part be temperature) will fieldnetic (e.g.,param unknown field intensity)magnetic or and potential magnetic vector a electromag of the parameters away such unknown in the of equations that governing a set of formalization the coupling should applied, method be direct the requiring cases, these In very effective.and However, approach could lead this some cases, to appreciable in errors. take place. will field andheat sources the electromagnetic of and satisfied,recalculation be Electromagnetic field distribution in the air, in most cases of coil design and process and process air, of thecoil design in cases most in fielddistribution Electromagnetic anetwork shown above,As modeling mesh of the require methods FDM FEM and both choice of the mesh generation in the only and Therefore, different are FDM FEM and overlap stencils of complex case difference one the another,Finite in workpiece and Superficially,andFDMFEM thedifferent; however, be to appear closely related.are they However, not well simulation for FEM of suited as as complex the usually FDM is bound approach most popular valid the For coupling is being most applications, IH indirect an and Final Remarks Regarding Computer Modeling Comparison Numerical of Different Techniques r ), convergence the no longer condition will Handbook of Induction Heating ofInduction Handbook ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 ficulties include the following: the include ficulties application. IH dif Some of ofthe aparticular features important address certain cannot programs developers.thatmany generalized simplified is software Theby result tially mightrelated be either IH to overlooked specifically subtleties process or substan certain base. abroad industrial within couldsimulation that used tools be possible developers as customers forced many to early as software to produce universal products were and systems later their adapted torecording sell need The to needs. IH magnetic and nondestructive motors, testing, breakers, transformers, machines, circuit place electrical in developed taking were that processes for originally modeling grams pro for computer codes used all-purpose of of are majority IH commercial modeling The 3.4.4 advantages provides applicationmethods treats better. substantial and specifics FDMmight have over superior of acombination different cases, qualities many FEM. In shaped bodies of classically case the problems. in heat and Only transfer electromagnetic of choice modeling most for popular the mathematical is method piece shape variation, this suited to computeFEMthe to of greater profiles. temperature work of Because the flexibility always are heated of are the welland parts FDMFEM defined, well boundaries both the application specific and onnique the depends subtleties. process right choice of The of simulation tech acombination methods. to it use cases, effective is advantages. very simply. many rized In certain has methods above-described of Each the mutual inductances. mutual and self-inductances of calculating limitations known of the because shaped bodies short execution time. mesh generationsystem, the it procedure relatively makes simple, fast and in resulting components of the induction of the boundaries of the only discretization BEMand require advantage over considered an be as methods can FDM FEM. MIM of and Since these This elements.nite However, shortcomings. abovementioned of certain each the has methods infi method, “mapping”and elements of combination finite and technique, “ballooning” the are methods Somethose exteriorregion. of phenomenon the addressing infinite of an wave of electromagnetic nature propagation. have Severalused, infinite methods been the with deals This putation to infinity. extends how exterior that region is an to treat when modeling elongatedwhen modeling systems. disadvantagebe consideredandFDMFEM,in the a particular can both of air the fieldin heater. induction the out to always need computation The carry electromagnetic of the from fieldexposure stage design when electromagnetic evaluating final the during only Background Theoretical Regardless of well-recognized impressive capabilities of modern commercial software, impressive software, capabilities of commercial of modern well-recognized Regardless Since simulation of as electromagnetics. not cumbersome as usually is modeling Thermal An introduction into numerical methods used for the simulation of IH can be summa be simulation for of can IH the used methods numerical into introduction An for complex- computational not effective does appear technique MIM to an be The consideration. into air the BEMs taking and do not impedance mutual require the Both Another difficulty that appears when using FDM or FEM for electromagnetic fieldFDMthat com FEMusing or for when appears electromagnetic difficulty Another • • induction coil(s).induction A heated workpiece simultaneously move, can to the rotate respect or, in oscillate (unsteady) radiation factors view consideration. into state thermal and its transient to take necessity the and refractory of presence a thermal The Limitations of Generalized All-Purpose Commercial Programs Commercial of GeneralizedLimitations All-Purpose 133 ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 ware preferred. are or soft methods numerical problems, code. certain of For similar universal afamily for a single searching rather than software application-oriented effective highly and of it processes, IH of preferable a wide is subtleties variety and to have of a number applications.inductionintothermal considerationall fits specifics Taking optimally simulations. of usefulness and the accuracy affect mightdramatically that notassumptions well-defined to make analyst an forcing software, all-purpose factor for particular cation could alimiting be an obstacle when using generalized commercial modeling software. modeling commercial generalized obstacle when using an Unexpectedly, of any induction forge feature design a common such heater become might place heat to surface (e.g., owing losses taking radiation). heat convection thermal and condition boundary on abillet’s temperature of refractory thermal effect the quantify shoulduser somehowthe that suggests manual The design. refractory of a specifics ting sate for the differences in refractory temperature. refractory in sate differences for the whatand would most appropriate the be would that recipe process allow you to compen conditions thermal of on billet versus acold ahot effect start the start to predict software system, induction of to the able and, be therefore, purchased geometry to expect the use process. subsequent heating the affects that cooling refractory machines). leads operation to partial Short interruption in of forming short interruptions cycle (e.g., process the in ruption related to issues of technological temporarily because ambient temperature (such alongambient temperature weekend). after as cooledwas refractory to down thermal and for off switched a sufficientlyits time long heater induction the was which conditionforging. aprocess in Cold represents start late stages (cold process polar two before hot heating and start) billet induction of the start 134 Our experience shows that there is not a single universal computational that method not shows universal is asingle experience there that Our Therefore, it is important to be aware that some critical feature(s)Therefore, to aware appli be IH some critical it that important is of aparticular Suddenly, you might realize that the purchased software package not does Suddenly, input allow software purchased the that you realize might Suppose you physical the refractory’sknow of and the properties its material, thickness, In contrast, hot start designates a condition in which there was arelatively there which contrast, acondition designates In in short inter hot start Imagine, for instance, that you have purchased software and make an attempt to an simu make and you that software for have instance, Imagine, purchased • • • • • • The presence of presence and end liners, plates,shields, The others. guides, fixtures, on. so components, and of probability cracking, shape/size residualphenomena, and stresses, transient of heat distortion treated viscoplastic and viscoelastic thermal transformations, phase as such nature, cal be ableto of a metallurgi to simulate not processes necessary sufficient. also is It phenomena heat is to simulateand coupled only transfer ability electromagnetic appreciable of an induction heat-treating is applications family There where the on. so and stage, place: multicoiltake cold holding, start, coil-emptying intermediate start, For example, bars, when or plates, billets, heating stages following transient the of presence not steady-stateThe only well. stages as process but cycles transient casting). continuous (i.e., distributions temperature after reheating initial of nonuniform existence The heating). bar in-line or dual frequency contourgear hardening (i.e., frequencies different of substantially two use or sequential Simultaneous hardening). stages (i.e., quenching and Some heating operations combine scan induction Handbook of Induction Heating ofInduction Handbook ------Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 Advanced Materials & Processes & Materials components,treated Advanced V. (From of heat groove. and prevent failures Rudnev, helps changes Computer modeling diameter having shaft ahollow hardening scan of induction dynamics sequential of the simulation computer of FEA Results FIGURE 3.65 begins appreciably in front of the copper turn, creating a preheating effect. Axial heat Axial effect. apreheating creating appreciably copper of front turn, the in begins wouldimpossible,if that [142].not experimentally difficult, be to determine comprehensiveprocess of of the induction equipment to details determine manufacturers shaft’s the geometry. in to accommodate computer Numerical changes allow simulations rate power coil and scan scanning The were during varied scanning. tor during positions provided, are induc at heat well shaft different as as of the distribution areas selected four Temperature analysis. was FEA modeled at using top half the variations only symmetrical, is groove.shaft the snap ring Because and changes diameter has that a hollow shaft ening hard of scan induction shows dynamics of computer results the sequential the modeling most application. appropriate is that to technique aparticular the to select allows them professional activity. their into computer-modeling This proprietary and techniques cial Background Theoretical As can be noted from Figure 3.65, during scanning, considerable shaft of 3.65, the heating Figure noted be from can scanning, As during StudyCase As a result, Inductoheat scientists and engineers utilize and integrate both commer integrate both and utilize aresult,As engineers Inductoheat and scientists . Computer modeling of induction scan hardening. As an example, an As . Computer Figure 3.65 hardening. of scan induction modeling 0.01 0.02 0.03 0.01 0.02 0.03 0.01 0.02 0.03 R R R 0 0 0 0 0 0 N1 =860.5N2699.8N3128.3N483.7 N1 =69.4N258.8N3143.2N4503.9 T N1 =276.4N2115.9N339.9N430.5 T ime =3.50(s) ime =9.00(s) T ime =18.20(s) 0.02 0.02 0.02 1 1 1 0.04 0.04 0.04 2 2 2 , October, 2011,, October, pp. 6–11.) O. 0.06 0.06 0.06 4 3 I.D. 4 3 4 3 D. Axis ofsymmetr 0.08 0.08 0.08 Gro ove y 0.1 0.1 0.1 Legend, °C 0 110 220 330 440 550 660 770 880 990 1100 135 - - - Downloaded By: 10.3.98.104 At: 18:13 28 Sep 2021; For: 9781315117485, chapter3, 10.1201/9781315117485-3 the the change. of adiameter regions the pronounced in inductor. scan the It particularly behind is shaft of the regions subsurface in accumulation failure. undesirable, to often apremature leading are potentially Such microstructures productsof (e.g., upper transformation or “ghost” bainitic/pearlitic networking). structures presence the with structures formation of the mixed in resulting curve, transformation cooling “nose” the create for conditions continuous of crossing potentially the and severity quenching reduce can This shaft. of surface where spray the on the the regions impinges in quenchant inductor, the even some behind cases, located and, immediately in areas of shaft postheating phenomenon. factor responsible another for is this coil fieldpropagation causing heat magnetic generation external ofoutsideinduction the the twoconductionis one flow of thermal factorsresponsible result as a of forThe preheating. 136 3.4.5 physicalerly all-important phenomena. modeling it capable and is of restrictions prop of critical it that free is sure make software, chasing and experience. 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In geometry Crucial Tips Executives Must Know Regarding Computer Modeling of IH Modeling Computer Regarding Know Crucial Executives Must Tips temperature. At the same time, the heat accumulated in the subsurface regions regions subsurface heat the temperature. the time, At in accumulated same the where you are seeking help. 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