LICENTIATE T H E S I S

Luleå University of TechnologyDepartment of Applied Physics and Mechanical Engineering

Division of Material Mechanics

2008:42|: 402-757|: -c -- 08 ⁄42 --

2008:42

Simulation of Induction Heating in Manufacturing

Universitetstryckeriet, Luleå

Martin Fisk

Simulation of Induction Heating in

Manufacturing

Martin Fisk

Abstract

Induction heating has been used during the past three decades in the industry.Because of fast heating and good reproducibility it is used in heat treatmentapplications, in bulk heating processed, as well as for special applications inthe food and chemical industry. Local heat treatmend by induction on repairwelded aerospace components is one example of induction heating in manu-facturing. Numerical simulation of induction heating enables optimisation ofthe process variables. Electromagnetic-thermal modelling of induction heatingusing finite element method is presented in this thesis work.

To be able to compute the temperature history within tight tolerances,an accurate model is needed. Well controlled and documented test-cases ofelectromagnetic induction heating are few in the literature. The first partof this thesis was therefore done to validate the induction heating model. Thework was then extended to include simulation of a manufacturing process chain.

iii

Acknowledgment

This work has been carried out at the division of Material Mechanics at LuleaUniversity of Technology from May 2006 to December 2008. The financialsupport was provided by the National Aviation Research Programme (NFFP1)in cooperation with Volvo Aero and University West.

First and foremost I would like to express my sincere gratitude to my supervisor,Professor Lars-Erik Lindgren for his enthusiasm and support during the courseof this work. I would also like to express my gratitude to my co-supervisor, Pro-fessor Hans O Akerstedt for sharing his deep knowledge in electromagnetism.

Many thanks to all my colleges at the University for making the work moreinspiring, especially those working in my division, Material Mechanics.

Finally I would like to thank Karin for support and putting up for me. Thankyou for being there for me.

1Nationella flygforskningsprogrammet

v

Dissertation

The thesis consists of an introduction part followed by two appended papers.The first sections are intended to introduce the reader to the basics of inductionheating and its opportunities. The papers are thereafter briefly commented asthe full description are available in the appended papers.

Paper A

Validation of induction heating model for Inconel 718 components.M. Fisk.Submitted to Journal of Manufacturing Science and Engineering.

Paper B

Simulations and measurements of combined induction heating and extrusionprocesses.S. Hansson, M. FiskSubmitted to Finite Elements in Analysis and Design.

vii

Contents

1 Introduction 1

1.1 Aim and scope . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The induction heating process . . . . . . . . . . . . . . . . . . . 1

2 Magnetic material properties 2

2.1 Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Cooperative magnetism . . . . . . . . . . . . . . . . . . . . . . 3

2.2.1 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . 32.2.2 Antiferromagnetism and ferrimagnetism . . . . . . . . . 3

2.3 Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.4.1 Temperature influence . . . . . . . . . . . . . . . . . . . 52.4.2 The Frolich representation . . . . . . . . . . . . . . . . . 7

3 Constitutive relations 9

3.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Constitutive relations . . . . . . . . . . . . . . . . . . . . . . . 103.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 113.4 The Poynting vector . . . . . . . . . . . . . . . . . . . . . . . . 123.5 The diffusion equation . . . . . . . . . . . . . . . . . . . . . . . 133.6 Skin depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.7 Linearisation of the permeability . . . . . . . . . . . . . . . . . 153.8 Units in electromagnetic fields . . . . . . . . . . . . . . . . . . . 16

4 Heat transfer 18

4.1 Heat transfer modes . . . . . . . . . . . . . . . . . . . . . . . . 184.1.1 Heat conduction . . . . . . . . . . . . . . . . . . . . . . 184.1.2 Convection . . . . . . . . . . . . . . . . . . . . . . . . . 184.1.3 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2 The heat conduction equation . . . . . . . . . . . . . . . . . . . 20

5 FE-solutions and summary of papers 21

5.1 Paper A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Paper B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Discussions and future work 25

ix

1 INTRODUCTION

1 Introduction

Welding changes the microstructure of the material thereby deteriorating themechanical properties of the component. Today, local heat treatment by elec-tromagnetic induction is used at Volvo Aero Corporation on repair welded partsto restore the microstructure. For large components, local heat treatment isfavoured over global, due to economic and practical reasons.

Induction heating is a convenient method useful for local heat treatment.The current work is aimed to enable modelling of this process and will inturn be useful when defining individual parameters affecting the temperaturedistribution. The temperature distribution in the workpiece depends primarilyon parameters like, coil position, electrical current through the induction coil,frequency of the current, thermal properties of the workpiece etc.

1.1 Aim and scope

This work is part of a project aimed to perform finite element simulation ofmanufacturing chain. Using the finite element method, it is possible to computethe temperature distribution in a component which is subjected to inductionheating. It is then possible to optimise the various process parameters. Thegoal of this work is to develop a validated model for induction heating analysis.This can be phrased as the following research question;

How should the induction heating process be modelled?

The approach taken to develop this model is to start with a simple geometryhaving all the essential features. The model is later extended to more complexindustrial cases where the temperature is mapped from the induction heatingmodel to subsequent models in a manufacturing chain. A commercial FE-software extended with user sub routines has been used for the papers that areappended in this thesis.

1.2 The induction heating process

Induction heating is a non-contact heating process used when heating an elec-trically conducting material by electromagnetic induction. In a thin layer of theworkpiece, usually called the skin depth, the alternating electromagnetic fieldgenerates a current which is called eddy current. The eddy current generatesheat, due to ohmic power losses and is the main heat source in an inductionheating process. In a ferromagnetic material the hysteresis effect does alsocontribute to heat generation, but in a much smaller amount.

The skin depth is defined as the depth at which the magnitude of the cur-rents drops e−1 from its surface. Its depth depends on the electric conductivityof the material, the frequency of the applied electromagnetic field and the mag-netic properties of the workpiece itself.

1

2 MAGNETIC MATERIAL PROPERTIES

2 Magnetic material properties

Each electron in an atom has a net magnetic moment originating from twosources, orbiting and spin. The first source is related to the orbiting motionof the electron around its nucleus. The orbiting motion of the electron giverise to a current loop, generating a very small magnetic field with its momentthrough its axis of rotation. The second magnetic moment is caused by theelectron’s spin and can either be in positive (up) direction or in negative (down)direction. Thus each electron in an atom has a small permanent orbital andspin magnetic moment.

From the simplified Bohr atomic model, the electrons of an atom are locatedin shells, denoted as K, L, M, and subshells (s, p, d), which are orbiting arounda nucleus. Each subshell has different energy states, consisting of electron pairs,with opposite spins cancelling each others magnetic moment. This cancellationalso hold for the orbiting electron. Therefor, for atoms with incomplete electronshells or sub shells, a net magnetic moments will occur.

When an external magnetic field, H, is applied on the body, the magneticmoment in the material tends to become aligned with this field, resulting ina magnetization M of the solid. Assuming a linear relation between the mag-netization vector and the magnetic field makes it possible to write [12, 14, 16]

M = χmH + M0 (1)

where χm is a dimensionless quantity called the magnetic susceptibility. Itis more or less a measure of how sensitive (or susceptible) a material is to amagnetic field. M0 is a fixed vector that bears no functional relationship to H

and is referred to the state of permanent magnetization.Let the magnetic flux density, B, represent the internal field strength within

a substance subjected to a external magnetic field H. Per definition, the mag-netic flux density and the magnetic field are related to each other according to

B = µ0(H + M) (2)

Combining Eq. (1) and (2) gives

B = µ0(1 + χm)H + µ0M0 = µ0µrH + µ0M0 = µH + µ0M0 (3)

The quantity µ is known as the permeability of the material and µr iscalled the relative permeability, which is the ratio of the permeability of agiven material to that of free space, µ0. The relative permeability of a materialis therefore a measure of how easily a material can be magnetized, or how muchbetter it conducts magnetic flux than in free space.

2.1 Paramagnetism

For some solid materials, the atoms have a permanent dipole moment due toincomplete cancellation of the electron spin and/or orbital magnetic moments.

2

2 MAGNETIC MATERIAL PROPERTIES 2.2 Cooperative magnetism

However, due to the thermal agitation, the atoms are randomly oriented andresult in zero net magnetic moment of the solid. Applying an external field onsuch material the atoms start to align, resulting in a small positive susceptibilityusually in the order of 10−5 to 10−2. Aluminum, titanium and tungsten areexamples of paramagnetic materials. Due their low relative permeability, µr,paramagnetic materials are all treated as nonmagnetic. See Fig. 1 for anillustration of the orientation of magnetic moments in a paramagnetic material.

Figure 1: Schematic illustration of the alignment of magnetic moments fordifferent types of materials.

2.2 Cooperative magnetism

There is a number of materials with atoms possessing permanent magneticmoment, as in paramagnetic materials, but with the difference that the atomscooperate with each other. Ferromagnetism, antiferromagnetism and ferrimag-netism are members of this family. Figure 1 shows how their magnetic momentsare aligned to each other.

2.2.1 Ferromagnetism

Ferromagnetic materials have a permanent magnetization even in the absenceof a magnetic field, and is due to the same net atomic magnetic moment asin paramagnetism i.e. electron spins that do not cancel. The orbital magneticmoment has also a contribution but is small compared to the spin moment.The difference with respect to a paramagnetic material is that the atoms arenot randomly oriented; they are interacting with each other in domains. Seesection 2.3. This results in a magnetic susceptibility which is in several ordersof magnitude larger than a paramagnetic material obsess. Susceptibility valuesas high as 106 are possible for ferromagnetic materials where cobalt, nickel andiron (BCC structure) are examples of ferromagnetic materials. Annealed agedInconel 718 is paramagnetic at room temperature.

2.2.2 Antiferromagnetism and ferrimagnetism

The magnetic moment of an atom in a antiferromagnetic material are antipar-allel to the neighboring atoms, resulting in zero net magnetization. Nickel

3

2.3 Domains 2 MAGNETIC MATERIAL PROPERTIES

oxide (NiO), manganese oxide (MnO) and iron manganese alloy (FeMn) areexamples of this material group [16]. A material that possess such magneticproperties react as if they where paramagnetic.

Ferrimagnetism is a combination of ferromagnetism and antiferromagnetism.However, in ferrimagnetic materials the opposing moments are unequal instrength, resulting in a net magnetization. This phenomena can occur whenthe sublattices consist of different materials or ions, such as Fe2+ and Fe3+ inFe3O4 [16].

2.3 Domains

A magnetic domain is a region where the individual magnetic moments of theatoms are aligned with each another. A ferromagnetic or ferrimagnetic materialis composed of several domains, individually changing their alignment. In apolycrystalline specimen the domains do not correspond with the grain in thematerial as each grain can consist of more than a single domain. The magnitudeof the magnetization, M, for the entire solid is therefore the vector sum of themagnetization for all the domains.

In a permanent magnet, such as a magnet that hold notes on a refrigeratordoor, the domains stays aligned and the vector sum is non-zero even in theabsence of a magnetic field. Materials that obsess this possibility is called hardmagnetic material. On the other hand, soft magnetic materials are materialsthat lose their memory of previous magnetization and can therefore not be apermanent magnetic.

2.4 Hysteresis

When a ferromagnetic or ferrimagnetic material is exposed to an externallyapplied magnetic field H, the relationship to the magnetic flux density B maynot be linear as in Eq. (3). A typical hysteresis curve showing a non-linearrelation between B and H and can be seen in Fig. 2.

Initially, domains in the unmagnetized specimen are in different directions.When an external magnetic field are applied on the specimen, the domains startto line up in the same direction as the applied magnetic field. This results inthe magnetization of the specimen. See Fig. 2a line 1. This orientation processwill continue until the saturation point Bs is reached and all domains are linedup. After this point there is a linear relation between changes in the magneticflux and the magnetic field.

When the magnetic field is reduced, it will not follow the initial curve butlags behind, i.e. line 2 in Fig. 2a. This phenomena is called the hysteresis.Whenever the magnetic field is zero, B is not reduced to zero but to Br. Thispoint is called the remanence, or remanent flux density; the material remainsmagnetized in the absence of an external field.

4

2 MAGNETIC MATERIAL PROPERTIES 2.4 Hysteresis

(a) A virgin hysteresis curve (b) A full hysteresis curve

Figure 2: When a specimen initially is magnetized, it follows the virgin curvedenoted 1 in Fig. a). If the magnetic field then is reduced it follows the uppercurve, denoted 2, and a residual magnetization Br remains if the magneticfield is reduced to zero. Fig. b) shows a full magnetization cycle where Hc

is the coercive force giving how strong the opposite magnetic field must be todemagnetizes a material.

To reduce the magnetic flux in the specimen to zero, a magnetic field of themagnitude Hc in the opposite direction (to the original one) must be applied.See Fig. 2b. Hc is called the coercive force.

The size and shape of the hysteresis curve is of practical importance. Thearea within a loop represent is the energy loss per magnetization cycle whichappear as heat that is generated within the body. This means that a hard ferro-magnetic material can have a heat effect due to hysteresis losses. Nevertheless,Rudnev et. al. [11] states that in a great majority of induction heating appli-cations, the heat effect does not exceed 7% compared to the heat generated bythe eddy current.

2.4.1 Temperature influence

When the temperature increases in a ferromagnetic or ferrimagnetic material,the ability to magnetize decreases. This is due to the increase in magnitudeof the thermal vibrations of the atoms, which tend to randomize any magneticmoment resulting in decrease of the magnetization.

The saturation point Bs has its maximum at 0K and decreases with increas-ing temperature. For each material there is a temperature where the ability to

5

2.4 Hysteresis 2 MAGNETIC MATERIAL PROPERTIES

magnetize abruptly drops, and the magnetic characteristics vanish. It becomesparamagnetic. This temperature is called the Curie temperature, Tc

2, for fer-romagnetic and ferrimagnetic materials. For antiferromagnetic materials thistemperature is called the Neel3 temperature.

The Curie temperature varies from different materials; for example, for iron,cobalt and nickel Tc are 768, 1120 and 335 C respectively. The phenomenacan be seen as a peak in the specific heat capacity curve of the material itself.The peak is at approximately 500 C for SAF 2507, one of the materials inpaper B, and is therefore its Curie temperature.

In Fig. 3 the hysteresis curves for SAF 2507 at different temperatures areshown. It can be seen that the curve above the Curie temperature is linearand have the same slope as µ0 over the whole range of applied magnetic field.This is also true when the magnetic flux has reached its saturation point andthe only contribution is from the linear relation between the magnetic flux andfield, i.e. µr = 1. SAF 2507 is a soft magnetic material, since the residual flux,Br, is close to zero.

−5 −4 −3 −2 −1 0 1 2 3 4 5

x 105

−1.5

−1

−0.5

0

0.5

1

1.5

H [A/m]

B [V

s/m

2 ]

20100200300400450480490500520600

Figure 3: The hysteresis curve for SAF 2507.

2After Pierre Curie.3After Louis Neel

6

2 MAGNETIC MATERIAL PROPERTIES 2.4 Hysteresis

2.4.2 The Frolich representation

There are a number of equations that mathematically can describe the hystere-sis curve. One example is the Frolich representation

B =H

α + β||H|| (4)

which is a good compromise between accuracy and simplicity [13, 9]. Knowinghow the variables α and β changes with the temperature, it is possible tocalculate the hysteresis curve for different temperatures. In Fig. 4 the variationof α and β versus temperature for SAF 2507 is shown. When the materialbecomes paramagnetic α tends to 1/µ0 = 1/(4π · 10−7) and β tends to zero.

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Temperature [°C]

β α

0

1

2

3

4

5

6

7

8x 10

5

Figure 4: The variation of α and β in Eq. (4) versus temperature for SAF2507.

The Frolich curves using the calibrated α and β values are compared withthe measured in Fig. 5 for SAF 2507. The used values for α and β give a goodagreement. Since there is a state of permanent magnetization in the material,an initial magnetic flux B0 is added for the temperature of interest to theFrolich equation.

7

2.4 Hysteresis 2 MAGNETIC MATERIAL PROPERTIES

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 105

0

0.2

0.4

0.6

0.8

1

1.2

1.4

H [A/m]

B [V

s/m

2 ]

20400450480490600

Figure 5: Calculated (lines) and measured (lines with stars) hysteresis curvesfor SAF 2507 at different temperatures. The legend gives the temperature inCelsius.

8

3 CONSTITUTIVE RELATIONS

3 Constitutive relations and basic concepts

The basic equations and their simplification, used in the finite element code,are summarised below.

3.1 Maxwell’s equations

Maxwell’s equations are a set of equations that describe the electric and mag-netic field and relate them to their sources, charge density and the vectorcurrent density. The set consists of Faraday’s law and Ampere’s circuital lawwith Maxwell’s extension. The extension term is also called the displacementcurrent. The equations are written as

∇× E = −∂B

∂t(5)

∇× H = J +∂D

∂t(6)

They consist of four different field variables; the electric field intensity E, themagnetic flux density B, the magnetic field intensity H and the electric fluxdensity D.

The differential equation

∇ · J = −∂ρ

∂t(7)

expresses the conservation of charges at any point. It is frequently referred toas the equation of continuity where ρ is the free volume charge density and J

is the current density [8, 14].Two further conditions can be deduced directly from Maxwell’s equations.

The divergence of Eq. (5) leads to

∇ · ∂B

∂t=

∂

∂t∇ · B = 0 (8)

because the divergence of the curl of any vector field is identically zero. Itfollows from Eq. (8) that at every point in the field the divergence is constant.If the field sometime in its past history has vanished, the constant must be zeroand the magnetic flux becomes solenoidal, i.e. [8, 14]

∇ · B = 0 (9)

which sometime is called Gauss’s law for magnetism.Similarly, the divergence of Eq. (6) leads to

∇ · J +∂

∂t∇ · D = 0 (10)

9

3.2 Constitutive relations 3 CONSTITUTIVE RELATIONS

and if it is assume that the field sometime in its past history has vanished

∇ · D = ρ (11)

which is known as Gauss’s law. Equation (9) and (11) are frequently includedas a part of Maxwell’s equations.

Among these five equations that are included in Maxwell’s equations, onlythree of them are independent. Either Eq. (5) - (7), or Eq. (5), (6) and (11)can be chosen as independent. The other two equations can be derived fromthese three, and are therefore called auxiliary or dependent equations [7].

3.2 Constitutive relations

The relation between the magnetic flux and field intensity was given in Eq.(3). A similar relation can be formulated between the electric flux and fieldintensity. Therefore,

D = ε0(1 + χe)E = ε0εrE = εE (12)

and (Eq. (3) again)

B = µ0(1 + χm)H = µ0µrH = µH (13)

where ε is the permittivity of the dielectric and χe is the electric susceptibilityof the material respectively. In a electric conductive material, the permittivityare usually set to one [12]. The constant values of permeability and permittivityof free space4 are known; µ0 = 4π · 10−7 Vs/Am and ε0 ≈ 8.854 · 10−12 As/Vmand are related to the speed of light in free space

c0 =1√

µ0 ε0

(14)

There is also a relation between the current density and the electric fielddensity such that

J = σE (15)

where σ it the conductivity of the material. It is the continuum form of Ohm’slaw.

If the medium is homogeneous then Eq. (11) can be written as

∇ · E =ρ

ε(16)

Combining the equation of continuity, Eq. (7), and Ohm’s law, Eq. (15), wecan write

∇ · σE +∂ρ

∂t= 0 (17)

4A concept of electromagnetic theory, corresponding to a theoretically ”perfect” vacuum,and sometimes referred to as the vacuum of free space.

10

3 CONSTITUTIVE RELATIONS 3.3 Boundary conditions

and with Eq. (16)

∂ρ

∂t+ σ

ρ

ε= 0 (18)

The charge density of any time is therefore

ρ = ρ0e−τt (19)

where τ = ε/σ is the relaxation time of the medium and ρ0 is equal to thecharge density at time t = 0. The initial charge distribution will thereforedecay exponential with a time equal to the relaxation time. A typical con-ductivity for a metal is of the order 106 Siemens/m with a permittivity ε of10−11 As/Vm. Consequently τ is of the order 10−17 s, resulting in that thecharges will vanish from an interior point of a metal body and appear at thesurface almost immediately. Even for such a poor conductor as distilled waterthe relaxation time is not greater than 10−6 s [14]. Even at high frequencies nocharge will be accumulated inside the conductor and Eq. (7) can therefore beapproximated as

∇ · J = 0 (20)

This allows an important simplification of Ampere’s circuital law with Maxwell’sextension. Assume a time harmonic field with an angular frequency of ω. Equa-tion (6) will then have a magnitude of the current and displacement currentas J and ωǫ/σ, i.e. 1 and ωτ . Therefore, a reasonable assumption is that thedisplacement current is negligible in a conductor even for high frequencies andEq. (6) can be written as [6],

∇× H = J (21)

This is consistent with Eq. (20), since the divergence of a curl is zero. Thisimportant simplification makes it possible to derive the diffusion equation forelectromagnetic fields. See section 3.5 for this derivation and further details.

3.3 Boundary conditions

Depending on the electric and magnetic properties, the electromagnetic fieldscan be discontinuous or continuous on each side of a common interface betweentwo different materials. A derivation on the boundary condition can be foundin for example Sadiku [12] or Stratton [14].

At an interface between two mediums the field must satisfy the followingconditions

n × (E1 − E2) = 0 (22a)

n · (D1 − D2) = ρ (22b)

n × (H1 − H2) = Js (22c)

n · (B1 − B2) = 0 (22d)

11

3.4 The Poynting vector 3 CONSTITUTIVE RELATIONS

where the boundary outward unit normal, n, is directed from medium 2 to-wards medium 1. The boundary conditions can also be formulated in wordsas: The electric field tangential components are continuous across the inter-

face of medium 1 and 2. The electric flux normal component is discontinuous across the interfacingmediums with a magnitude of ρ. Note that for a electrically conductivemedium, ρ = 0. The tangential components of the magnetic field strength are discontin-uous across the two mediums with a magnitude of Js. In the case of azero surface current, as when the medium has a finite conductivity, thetangential component is continuous [8, 14]. The magnetic flux density normal component is continuous across theinterface of medium 1 and 2.

3.4 The Poynting vector

Poynting’s theorem is a relation between the rate of change of energy storedin the fields and the energy flow [14]. Multiply Faraday’s law, Eq. (5), andAmpere’s law ignoring the displacement current, Eq. (6), with the electric andmagnetic field respectively, and subtract them from each other gives

∮

s

(E × H) · n dA = −∫

v

H · ∂B

∂tdv −

∫

v

σE2dv (23)

The quantity E × H is known as the Poynting vector P in watts per squaremeter. Poynting’s theorem states that the sum of the ohmic loss and the powerabsorbed by the magnetic field in the volume is equal the power input of thebody.

It is common to represent a harmonic variation in time as a complex functionof coordinates, multiplied by the factor eiωt. The complex Poynting vector cantherefore be written as [14, 13]

P =1

2(E × H∗) (24)

where (H∗) is the complex conjugate of the magnetic field. Equation (23) hasthen the complex form [13]

∮

s

P · dS = −j2ω

∫

v

1

4µ0µr|H|2dv −

∫

v

1

2σE2dv (25)

The real part of Eq. (25) determines the energy dissipated in heat (ohmiclosses) in the volume v and the imaginary part is equal to 2ω times the mean

12

3 CONSTITUTIVE RELATIONS 3.5 The diffusion equation

values of magnetic energy. The current from the real part is known as the eddycurrent5 and is the main heat source in induction heating.

3.5 The diffusion equation

To describe a vector field completely, both the divergence and curl have to beuniquely defined. Since the magnetic flux density satisfies a zero divergencecondition, it can be represented as the curl of another vector such that

B = ∇× A (26)

where A is the magnetic vector potential. From Maxwell’s equations it is knownthat

∇× E = −∂B

∂t= − ∂

∂t(∇× A) = −∇× ∂A

∂t(27)

or

∇× (E +∂A

∂t) = 0 (28)

Because ∇× (∇ϕ) = 0, one solution to Eq. (28) is

E = −∂A

∂t−∇ϕ (29)

The negative sign in front of ∇ϕ is per definition [12]. Multiply the electricfield with σ

J = σE = −σ∂A

∂t− σ∇ϕ = −σ

∂A

∂t+ Js (30)

where Js is the source current density in the induction coil [11]. It follows fromEq. (21) that

∇× 1

µB = J (31)

Substitute Eq. (26) and (30) we obtain

∇×∇× A = −µσ∂A

∂t+ µJs (32)

for a homogeneous, isotropic medium independent of field intensity. Expandand impose on A the Coulomb gauge condition ∇ · A = 0 [7],

σ∂A

∂t− 1

µ∇2A = Js (33)

5Discovered by the French physicist Leon Foucault. Localised areas of turbulent waterknown as eddies give rise to the eddy current.

13

3.6 Skin depth 3 CONSTITUTIVE RELATIONS

which is called the diffusion equation. The equation can also be expressed inthe same form but for different vectors; instead of A we write J, E, H, B [6].In the case of the magnetic vector potential, it can be related to any physi-cally observable electromagnetic induction phenomenon such as eddy current,induced voltage, coil impedance, flaw impedance, coil inductance etc. [5]. Ifthe excitation current is assumed to be sinusoidal, and the eddy current as well,a time-harmonic electromagnetic field can be introduced in Eq. (33) and wegain

iωσA − 1

µ∇2A = Js (34)

It should be noted, that in a ferromagnetic material the harmonic approxi-mation is not valid anymore. This is due to its the time dependent relativepermeability µr, which make the eddy current non sinusoidal [9]. See section3.7 for how to approximate the behaviour of ferromagnetic materials, using thisharmonic approximation.

3.6 Skin depth

Maxwell’s equations can be expressed as two second order differential equationsin E and H alone. Assume that the material is linear, isotropic, homogeneousand charge free (ρ = 0). Start with taking curl on equation (5) and (6)

∇×∇× E = −µ∂(∇× H)

∂t= −µ

∂

∂t

(

σ + ε∂

∂t

)

E (35)

∇×∇× H =

(

σ + ε∂

∂t

)

∇× E = −µ∂

∂t

(

σ + ε∂

∂t

)

H (36)

Furthermore, harmonic solutions are assumed and the relation

∇×∇× a = ∇ (∇ · a) −∇2a (37)

is used together with the properties ∇ ·H = 0 and ∇ ·E = 0. These steps leadto two second order homogeneous wave equations;

∇2E − γ2E = 0 (38a)

∇2H − γ2H = 0 (38b)

where γ2 = iωµ(σ + iωε) and γ is called the propagation constant of themedium. Let the propagation constant be defined as

γ = α + iβ = iω

√

µε

(

1 − iσ

ωε

)

(39)

14

3 CONSTITUTIVE RELATIONS 3.7 Linearisation of the permeability

The solution of equation (38b) is, if it represents a uniform electric wave prop-agating in the positive z-direction [12],

Ex(z, t) = Re[E0e−αze−iβzeiωt] = E0e

−αzcos(ωt − βz) (40)

Consequently, if a electric or magnetic wave travels in a conductive medium, itsamplitude starts to attenuate exponentially by a factor e−αz. The factor α istherefore called the attenuation constant, with the unit Np/m (Neper/meter).

A good conductor is per definition if σ >> εω why Eq. (39) can be simplifiedto6

γ =√

i√

µωσ =1 + i√

2

√

µ2πfσ = (1 + i)√

πfµσ (41)

The skin depth is defined as the distance for which the amplitude of a planewave decreases a factor e−1 = 0.368. Thus it becomes

δ =1

α=

1√πfµσ

(42)

Therefore, when the electromagnetic waves reach the surface in a conductingmedium, a current will start to flow within the surface and decreasing expo-nentially towards the interior of the material. The total current in a conductoris given by [13],

I =

∫ 0

−∞

J dy =Js

α=

Js δ√2

e−i π

4 (43)

which is equal to the r.m.s.7 value of the surface current density flowing uni-formly in a layer of δ lagging the coil current (and the magnetic field) by 45.However, it is important to remember that the electromagnetic field existsbelow the skin dept.

3.7 Linearisation of the permeability

In nonlinear, like SAF 2507, materials Eq. (34) is not valid anymore. Thisis due to the time dependent relative permeability µr, which makes the eddycurrent non sinusoidal. There are two possibilities to overcome this; introducea complex permeability, or introduce a constant permeability. Labridis andDokopoulos [9] shows that it is possible to approximate the nonlinear materialwith a fictive linear material that has a constant unknown relative permeability,µf

r . This permeability is related to the non-linear B − H characteristic curveat every point. See Fig. 3. The condition that the linear fictitious material

6In polar form: i = ei

π

2 = ei

π

4 ei

π

4 = ( 1+i√

2)2

7Root mean square

15

3.8 Units in electromagnetic fields 3 CONSTITUTIVE RELATIONS

should have the same average heat density losses as the nonlinear material atevery point can be used to derive the fictive relative permeability as

µfri =

w1i + w2i

µ0 ∗ (Hfmi)

2(44)

where w1i and w2i is the magnetic co-energy density of the material. w1i isrelated to the actual B−H curve and w2i is related to the average value of theslope dB/dH during a quarter of a period T . Hf

mi is the maximum magneticfield intensity for a sinusoidal function in the fictitious material. The magneticco-energy density can be calculated as

w1i =Hf

mi

β− α

β2ln

α + βHfmi

α(45)

w2i =1

2

(Hfmi)

2

α + βHfmi

(46)

giving an over- and underestimation of heat losses respectively.Since both the the relative permeability of the fictitious material, µf

r , andthe magnetic field intensity, Hf

m, are unknown, an iterative process is necessary.In the first step µf

r is set to the initial slope of the Frolich curve, i.e.

µfr =

1

µ0α(47)

In all the next steps µfr will be calculated from Eq. (44).

The formulation is based on the Frolich representation of the nonlinearB − H curve. See Eq. (4) and section 2.4.2 for further information.

3.8 Units in electromagnetic fields

A new system of units was after Maxwell’s discovery proposed by Giorgi8 1901to the AEI9, since it was clear that electric measurements could not be ex-plained in terms of the three fundamental units of length, mass and time. Thenew system had the fundamental units metre, the kilogram and the second(m.k.s) and as a fourth unit any electrical quantity belonging to the practicalsystem such as the coulomb, the ampere, or the ohm. From the field equations,it is then possible to deduce the units of every field quantity in terms of thesefour fundamental units [14].

In a plenary session in 1935, the IEC10 adopted unanimously the m.k.ssystem, although the fourth unit was not chosen. Giorgi himself recommendedthe ohm to be the fourth unit, but it was not followed strictly. Fore example, in

8Giovanni Giorgi, an electrical engineer who invented the Giorgi system of measurement.9Associazione Elettrotecnica Italiana

10International Electrotechnical Commission

16

3 CONSTITUTIVE RELATIONS 3.8 Units in electromagnetic fields

the book written by Stratton in 1941 [14], the metre-kilogram-second-coulombsystem is used.

In 1960, at the 11th General Conference of Weights and Measures, theInternational System (SI) was adopted. It is bases on seven fundamental units;the metre, the kilogram, the second, the ampere, the kelvin, the mole and thecandela. In Table 1 the SI units and their related bas units are shown for themost common derived quantities in electromagnetism.

Table 1: Basic quantities in electromagnetism.

Derived quantity Symbol SI unit SI base unit

Electric field intensity E V/m mkg s−3 A−1

Electric flux density D C/m2 m−2 s AMagnetic field intensity H A/m m−1 AMagnetic flux density B T m2 kg s−2 A−1

Current Density J A/m2 m−2 AMagnetic flux φ Wb m2 kg s−2 A−1

Permeability µ H/m mkg s−2 A−2

Permittivity ε F/m m−3 kg−1s4 A2

Volume charge density ρ C/m3 m−3 s AConductivity σ S/m m−1kg−1s3A2

Resistivity ρc Ωm m3 kg s−3A−2

17

4 HEAT TRANSFER

4 Heat transfer

4.1 Heat transfer modes

There are three modes of heat transfer, conduction, convection and radiation.Heat transfer by conduction occurs inside the solid and the other modes areactive in the thermal boundary conditions.

4.1.1 Heat conduction

The empirical constitutive law for heat conduction is called the “Fourier lawof heat conduction” and is written as

qcond = −k∇T (48)

where k is the thermal conductivity, T is the temperature and qcond is the heatflux by conduction. The conductivity is usually a function of temperature.Typically is a uniform temperature distribution along the workpiece easier toobtain if the thermal conductivity is high.

4.1.2 Convection

Heat transferred by convection from the surface of the workpiece to the ambientfluid or gas can be expressed as

qconv = h(Ts − T∞)α (49)

where h is the convection surface heat transfer coefficient11, Ts is the surfacetemperature, T∞ is the ambient temperature and qconv is the heat flux denityby convection.

The value of the film coefficient depends primarily on the thermal proper-ties of the surrounding gas fluid, its viscosity and the velocity of the gas. Inmany induction heating applications, the workpiece is moving at high speed(e.g., heating of rotating disk, wire heating, etc.). Then the convection can beconsidered as forced. This convection can be equal or exceed heat losses due toheat radiation for low-temperature induction heat treatment. For a workpiecein open air a good approximation is [4, 11]

qconv = 1.54(Ts − T∞)1.33 (50)

which is compared with heat losses due to radiation in Fig. 6.

11Also known as the film coefficient.

18

4 HEAT TRANSFER 4.1 Heat transfer modes

4.1.3 Radiation

Heat losses transferred from the hot workpiece due to electromagnetic radiationis called thermal or heat radiation. The equation that describe this can beexpressed as

qrad = σε[(Ts)4 − (T∞)4] (51)

where σ is the Stefan-Boltzman constant (σ = 5.67 · 10−8W/m2K4) and ε isthe emissivity of the surface. The emissivity is defined as the ratio of the heatemitted by the surface to the heat emitted by a black body. Ts is the surfacetemperature and T∞ is the ambient temperature given in Kelvin.

Since the radiation loss is proportional to the fourth power of temperatureit exceeds the forced convection loss at elevated temperature, especially if theemissivity is larger than 0.3 as is seen in Fig. 6. Note that the value of theemissivity can vary for the same material. It varies from 0.03 for a for a polishedcopper surface to 0.70 for a heavily oxidised surface [4].

0 100 200 300 400 500 600 700 800 900 1000 11000

20

40

60

80

100

120

140

Temperature [°C]

Hea

t los

ses

[kW

/m2 ]

ε = 0.1q

convection

ε = 0.8

qradiation

Figure 6: The heat losses due to convection and radiation transfer modes. Thefull lines represent the radiation heat loss due to an emissivity varying from0.1 to 0.8 with a 0.1 increase for each line. The dashed line represents theconvection heat loss according to Eq. (50).

19

4.2 The heat conduction equation 4 HEAT TRANSFER

4.2 The heat conduction equation

The temperature distribution in a medium is governed by the heat transferequation [10]

ρc∂T

∂t−∇ · (k∇T ) = Q (52)

where ρ is the density, c is the specific heat capacity, k is the thermal conductiv-ity and Q the energy generated in the material per unit volume and time. Theheat transfer equation determines the temperature distribution in a medium asa function of space and time. Once the temperature distribution is known, theheat flux within the body or by its surface can be calculated from Eq. (48).

The solution of the heat conduction equation requires initial and boundaryconditions. The boundary condition can either be prescribed12, where thetemperature is known on the boundary, or/and as an imposed flux13 usingthe radiation and convection boundary. The flux condition is related on theboundary as

qn = −k∂T

∂n(53)

where n is the outward direction normal to the surface and qn the constantflux given as

qn = h(Ts − T∞)α + σε[T 4s − T 4

∞] (54)

if both convection and radiation boundary condition, as given in sections 4.1.2and 4.1.3, are assumed.

The similarity with the diffusion equation stated in chapter 3.5 can benoted. Suppose the temperature is varying on the surface of the material in asinusoidal mode. It can then be shown that the amplitude of the variation willbe smaller with increasing depth and with some time delay, i.e. a phase angle.If the frequency of the temperature variation is increased, the attenuation andthe phase angle will increase [6, 13]. This is the same phenomena as for theelectromagnetic wave case, but with another time scale, as the electromagneticfield has a much shorter relaxation time. See Carslaw and Jaeger chapter 2.6for the solution of the heat transfer equation due to sinusoidal heating of thesurface [1].

Even though the diffusion equation, Eq. (34), has the same properties asthe heat conduction equation, Eq. (52), their time scales differ. A harmonicapplied current in a induction heating equipment are usually in the frequencyrange of 50Hz − 70KHz and for wire up to 4000KHz [11]. Therefore, the useof Eq. (34) is possible within each time increment in the thermal analysis asthe time scale of the electromagnetic problem is much shorter. This is utilisedin the solution strategy described in section 5.

12The Dirichlet condition13The Neumann condition.

20

5 FE-SOLUTIONS AND SUMMARY OF PAPERS

5 FE-solutions and summary of papers

Practical problems governed by the diffusion and the heat conduction equationsare usually impossible to solve analytically, especially when the material hastemperature dependent and non-linear properties. A common approach toobtain solutions for such systems is to use the Finite Element Method. In thiswork the FE-software MSC.Marc has been used, allowing a coupling betweenthe harmonic electromagnetic solution and the heat transfer equation.

In order to calculate the temperature field within a body, a harmonicallyoscillating current is applied in the coil and from Eq. (34) the magnetic vec-tor potential is calculated. The generated current within the body, the eddycurrent, is then calculated from Eq. (29) as

Je = −iωσA (55)

The heat source term in the heat conduction equation, Eq. (52), can thenbe computed [15, 3] as

Q =1

2σ|Je|2 =

1

2σJe · J∗

e (56)

It should be noticed, that for all nodes where the source current is zero, ∇ϕ isset to zero. On the outer boundary of the region, the magnetic vector potentialis selected such that it is zero along the boundary (Dirichlet condition).

From Eq. (34) it is seen that the solution of the magnetic vector potentialis a steady state solution, using the material properties from the time step tn.This is possible since the time scale for the electromagnetic problem is muchshorter than in the thermal problem. Figure 7 shows the solution strategyduring a time step. The electromagnetic problem is firs solved using the elec-tromagnetic properties for the temperature at the time step Tn. Thereafter,the heat load due to the eddy current is calculated and the thermal problem issoled giving temperatures Tn+1 at the end of the time step.

21

5.1 Paper A 5 FE-SOLUTIONS AND SUMMARY OF PAPERS

Figure 7: The coupling between the electromagnetic and thermal problems

5.1 Paper A

The study done in this paper was aimed to provide a test case for validatingthe induction heating model. It is the first step in a larger project where man-ufacturing chains will be simulated. The test case consists of two experimentalarrangements; a cylinder heated along the circumference by a coil in the mid-dle, or at its end. See Fig. 8 for the magnetic field around the coil located inthe middle.

The published works in simulation of induction heating has been limited tokeeping the current constant in the coil during the heating stage, allowing thetemperature to vary [3, 15, 2]. However, here is the current is controlled inorder to obtain a particular temperature history at a specified location of theworkpiece.

The good agreement between simulations and measurements shows that themodel, the computational approach, as well as used material properties is validand can be used to study the induction heating process for Inconel 718. SeeFig. 9 for the temperature agreement when the coil was located in the middle.

22

5 FE-SOLUTIONS AND SUMMARY OF PAPERS 5.1 Paper A

Figure 8: The workpiece with coil and the magnetic field generated from thecurrent at 23 KHz.

0 100 200 300 400 500 600 7000

100

200

300

400

500

600

700

800

Time [s]

Tem

pera

ture

[o C]

Calculated dataMeasured data7

6

5

Figure 9: Measured and computed temperature for the coil located in themiddle of the workpiece. The numbers refers to positions of the thermo-couple.See Paper A for further information.

23

5.2 Paper B 5 FE-SOLUTIONS AND SUMMARY OF PAPERS

5.2 Paper B

Four steps in a manufacturing process of extruded tubes are studied. Two ofthe steps are induction heating and two of them are expansion and extrusion ofstainless steel tubes respectively. Axisymmetric FE-models of these steps, in-cluding cooling at the intermediate transports have been combined. All modelswere developed in the FE-software MSC.Marc and the nodal temperatures weretransferred from one model to another using an external mapping program.

Two different materials were used in the process flow simulation. One is anaustenitic stainless steel and the other is a duplex, austenitic/ferritic, stainlesssteel. Since the duplex steel is ferromagnetic below the Curie temperature,the skin effect is more pronounced compared to the paramagnetic austeniticmaterial. Also the magnetic non-linearity of the material properties of the bil-let leads to differences in the induction heating modelling. Results from thenumerical models have been validated by experiments in production pressesand induction furnaces. The result of the measure temperature and the calcu-lated i seen in Fig. 10. Since MSC.Masc uses a harmonic approximation, thepermeability has been linearised as is described in section 3.7.

0 100 200 300 400 500 600 700 8000

200

400

600

800

1000

1200

1400

Time [s]

Tem

pera

ture

[o C]

5 mm10 mm20 mm30 mm40 mm64 mm78 mm

Figure 10: The measured temperatures (dashed lines) compared with calcu-lated temperatures (full lines). The legend gives the depth in the workpiece atwhich the measurements has been carried out at.

24

6 DISCUSSIONS AND FUTURE WORK

6 Discussions and future work

How should the induction heating process be modelled? was the research ques-tion asked in the beginning of this thesis. The goal of this work has thereforebeen to develop a validated finite element model for simulation of the inductionheating process. In paper A, the temperatures as well as the current shows agood agreement between simulations and measurements. This shows that themodel and the computation approaches as well as used material properties arevalid and can be used to study the induction heating process.

The temperature agreement for the duplex steel in paper B shows that theapproach Labridis and Dokopoulos [9] proposed, to linearize the permeability,is a good approximation. This is important since a harmonic solution with aconstant permeability is much faster than an electromagnetic non-linear solu-tion [3]. In this work, four steps in a manufacturing process chain are simulated.The resulting temperatures from the electromagnetic-thermal calculation aremapped as an initial temperature to the thermo-mechanical calculation usingan external mapping tool.

In the future work, the simulations are extended to three dimensions, wherethe calculated temperature field are used in thermo-mechanical calculations.Welding changes the microstructure of the material and thereby deterioratingthe mechanical properties of the component. The microstructure needs dueto this be restored either by local or global heat treatment. The temperaturefield must then be mapped from the induction heating model to the thermo-mechanical model every time step.

25

REFERENCES REFERENCES

References

[1] H. S. Carslaw and J. C. Jaeger. Conduction of heat in solids. OxfordUniversity Press, Oxford, 2. ed. edition, 1959.

[2] C. Chaboudez, S. Clain, R. Glardon, D. Mari, J. Rappaz, andM. Swierkosz. Numerical modeling in induction heating for axisymmet-ric geometries. Magnetics, IEEE Transactions on, 33(1):739–745, 1997.0018-9464.

[3] S. Clain, J. Rappaz, J. Swierkosz, and R. Touzani. Numerical modelingof induction heating for two-dimensional geometries. Mathematicl Models

and Methods in Applied Sciences, 3(6):805–822, 1993.

[4] E. J. Davies. Conduction and induction heating. Number 0-86341-174-6in IEE Power Engeneering Series 11. Peter Peregrinus Ltd., London, UK,1990.

[5] C. V. Dodd and W. E. Deeds. Analytical solutions to eddy-current probe-coil problems. Journal of Applied Physics, 39(6):2829–2838, 1968.

[6] Percy Hammond. Applied electromagnetism. Pergamon Press, Oxford,1985.

[7] Jianming Jin. The Finite Element Method in Electromagnetics, volume 2.John Wiley & sons, inc., 2002.

[8] Gerhard Kristensson. Elektromagnetisk vagutbredning. Studentlitteratur,1999.

[9] Dimitris Labridis and Petros Dokopoulos. Calculation of eddy currentlosses in nonlinear ferromagneitc materials. IEEE Transactions on Mag-

netics, 25(3), May 1989.

[10] Roland W. Lewis, Perumal Nitiarasu, and Kankanhalli N. Seetharamu.Fundamentals of the finite element method for heat and fluid flow. Wiley,Chichester, 2004.

[11] Valery Rudnev, Don Loveless, Raymond Cook, and Michan Black. Hand-

book of Induction Heating. Inductoheat, Inc., 2003.

[12] Mattew N.O. Sadiku. Elements of Electromagnetics, volume 3. OxfordUniversity Press, Inc., 2001.

[13] Richard L. Stoll. The analysis of eddy currents. Oxford University Press,Oxford, 1974.

[14] Julius Adams Stratton. Electromagnetic theory. IEEE Press, Piscataway,NJ, 2007.

27

REFERENCES REFERENCES

[15] K.F. Wang, S. Chandrasekar, and Henry T.Y. Yang. Finite-element simu-lation of induction heat treatment. Journal of Materials Engineering and

Performance, 1, 1992.

[16] Jr. Willian D. Callister. Materials science and engineering: an introduc-

tion. John Wiley & Sons, Inc., 1999.

28

Paper A

Validation of induction heating model for Inconel718 components

Martin FiskDivision of Material Mechanics

Department of Applied Physics and Mechanical EngineeringLulea University of Technology

SE-971 87 Lulea, SwedenEmail: martin.fisk@ltu.se

Solving the coupled electromagnetic-thermal problemof induction heating requires simplifications in the nu-merical algorithm, mainly due to the different timescales of the fields. Temperature dependent mate-rial properties is another uncertain issue in modelling.However, well controlled and documented test cases arefew in the literature and therefore the current study wasdone as a first step in a larger project where manufac-turing chains will be simulated. The induction heatingprocess is used for local heat treatment of aerospacecomponents. This paper describes measurements andsimulations of induction heating of an axisymmetric rodmade of Inconel 718. The model was found to have agood accuracy. Furthermore, an approach combiningthe finite element code with a control algorithm of thecurrent has been implemented and is described. Thisalgorithm makes it possible to obtain the variation ofcurrent needed to create a given temperature history.

1 IntroductionThe basic setup of an induction heat equipment

consists of a coil with one or several windings, keep-ing an electrically conducting workpiece within. Aneddy current is generated in the workpiece as a conse-quence of the alternating electromagnetic field, and isproduced by the current carrying coil. In a thin sur-face layer of the workpiece (skin depth), heat is gener-ated due to the Joule heating effect caused by the eddycurrent. The skin depth depends on the frequency ofthe electromagnetic field and the material properties,resulting in changing temperature distribution for dif-ferent frequencies. Induction heating is a convenientnon-contact heating method, used in the industry dur-

ing the past three decades. Due to fast heating rate andgood reproducibility, it is used in applications such ashardening, tempering and annealing [1].

Welding changes the micro-structure on the mate-rial and a heat treatment is needed to restore it. Fur-thermore, it introduces residual stresses in the materialwhich will be reduced by the heat treatment. For largecomponents, local heat treatment by induction heatingis favoured over global, due to economic and practicalreasons. Temperature distribution in the workpiece de-pends on parameters like, coil position, electrical cur-rent through the induction coil, frequency of the cur-rent, thermal properties of the workpiece, as well asmany other factors.

The published works in simulation of inductionheating has been limited to keeping the current constantin the coil during the heating stage, allowing the tem-perature to vary [2–4]. However, in this work, currentis controlled in order to obtain a particular temperaturehistory at a specified location of the workpiece. Thismakes it possible to do a more traditional local heattreatment with the temperature kept constant under acertain time for full annealing processes or ageing [1].Without controlling the current, it is only possible totemper parts; self-tempering, where residual heat in thepart is used, and induction tempering, where the hard-ened part is reheated [1].

This paper describes measurements and finite ele-ment simulations of an axisymmetric rod made of In-conel 718. The computational model was validated bytwo experimental arrangements. A cylinder was heatedalong the circumference by a coil in the middle or atits end. The first case is similar to the setup in [3, 5].

Luo and Shih [5] perform inverse thermal analysis anddid not solve the electromagnetic problem. Therefore,their approach cannot be used to design general induc-tion heating setups. Wang et. al. [3] compare their finiteelement analysis with analytical solutions.

The finite element analysis was performed togetherwith a user routine; a logic for controlling the current inorder to produce specified temperature. The obtainedcurrent history agreed well with the measured current.The reverse calculation was also done where the mea-sured current was applied to the model and the com-puted temperature was found to agree well with themeasured.

Additionally, simulations where performed to eval-uate the sensitivity of the setup. They show that a smallchange in the current give relatively large variation inthe temperature of the workpiece. Frequency variationis also shown to affect the temperature.

2 The coupled electromagnetic-thermal problem

The basic equations and their simplification, usedin the finite element code, are summarised below.

2.1 Electromagnetism

The electromagnetic wave propagation is describedby Maxwell’s system of equations

∇×E = −∂B∂t

(1)

∇×H = J+∂D∂t

(2)

∇ ·B = 0 (3)

∇ ·D = ρv (4)

The constitutive relations between the previous men-tioned field quantities in an isotropic homogeneousmedium are

B = µH (5)

D = εE (6)

J = σE (7)

The displacement current is neglected and a harmonicsolution is assumed. The latter means that the excita-tion current and its solutions are sinusoidal functions.Furthermore, the magnetic vector potentialA is intro-

duced. It is related to the magnetic flux density by

B = ∇×A (8)

Maxwell’s equations can then be written so that

iωσA− 1µ

∇2A = J0 (9)

which is a complex diffusion equation. A similar equa-tion can also be formulated in terms of the electric fieldquantity [6,7].

The domain for Eq. (9) is the rod, coil and sur-rounding air. Additionally boundary conditions for theinterface between solids and air as well as externalboundary are required. Since there is no surface cur-rent in the electrically conducting material, the tangen-tial vector on the interface between the mediums arecontinuous which gives us [8]

n ×(E1−E2) = 0 (10)

n ×(H1−H2) = 0 (11)

where the boundary outward unit normal is directedfrom medium 2 towards medium 1. On the outer bound-ary of the region, the magnetic vector potential is se-lected such that it is zero along the boundary (Dirichletcondition).

2.2 Heat conduction

The heat conductivity in a solid is governed by

ρc∂T∂t

−∇ · (κ∇T) = Q (12)

The formulation accounts for temperature dependentmaterial properties. The solution of the equation re-quires initial and boundary conditions. A general modelfor the boundary conditions are

qn = h(Ts−Ta)+σsεw(T4s −T4

a ) (13)

which is related to the gradient of the temperature fieldin the normal direction of the surface,∂T/∂n.

In the case of a harmonic electromagnetic loading,

the heat source term can be written as [2,3],

Q =1

2σ|Je|2 =

12σ

Je ·J∗e (14)

whereJe is the induced current density in the conductor,calculated asJe = −iωσA. The heat source term,Q, isthen solved for each node point whenever Eq. (9) issolved, and used as input to Eq. (12) for a solution ofthe heat distribution. The use of Eq. (14) is possible asthe time scale of the electromagnetic problem is muchshorter than in the thermal problem.

3 Control logic for currentIn order to obtain a desired temperature, a Propor-

tional Integral Derivative (PID) controller is used toregulate the current in the computational model. Thedeviation between the prescribed and actual tempera-ture is the error e. This is used to compute a correctionu, to the current according to

u(t) = K

[

e(t)+1TI

Z t

0e(τ)dτ+TD

dedt

]

(15)

The algorithm is often illustrated by the block diagramin Fig. 1

Fig. 1. Block diagram of the PID controlling system. The process,

G, represents by the FE-program, e is the error and u the correction.

The discrete version of Eq. 15 is implemented as

un = K

[

en +∆tTI

n

∑i=0

ei +TD

∆t(en−en−1)

]

(16)

This is used to modify the current for the next increment

Pn+1 = Pn +un (17)

By adjusting the length of the time step and the different

constants the coil current is regulated giving an outputtemperature closed to the prescribed temperature.

4 Experimental setupThe induction heating equipment is a voltage-fed

series load with a capacitor at the input of the inverterand a series-connected output circuit. To transfer max-imum power from the DC-source to the load, the trans-former needs to work in its resonance frequency. There-fore during, the first 12 s of operation, the inductionheating equipment searches for its resonance frequency.See [1] for more information. The induction heatingequipment used in the experiments is Minac 18-5.

Fig. 2. The geometry of the workpiece and position of the thermo-

couples. See Table 1 for the distances. The coil has a dimension of

8x4 mm.

Table 1. Positions of thermo-couples.

Measurement Distance from Distance from

location center [mm] the end [mm]

1 0 0

2 8 0

3 16 1

4 16 12

5 16 39

6 16 60

7 16 70

Fig. 2 shows the experimental setup; the coil isplaced in position A (at the middle of the workpiece) orat B (at the end of the workpiece). The rod is 140 mm

long and has a diameter of 32 mm. The numbers in cir-cles denote the different thermo-couples with locationsgiven in Table 1.

The prescribed temperature history in the two dif-ferent cases consist of three phases:

1. Linear heating at a rate of (760/180) °C/s2. Temperature is held constant at 760 °C for 120 s.3. Current turned off, and the workpiece is allowed to

cool down to a uniform temperature.

The temperature was measured using thermo-couples and recorded at a frequency of two samples persecond. Each test was repeated three times to ensurethe reproducibility. In the following result figures, themean value of the temperature for the three tests areplotted. The maximum temperature difference betweenthe replicated tests was 7.3 °C at time 735 s for the coillocated at position A and 7.4 °C at time 176 s for thecoil at position B. The thermo-couples were calibratedbefore and after their use.

The difference did not exceed 1.1 °C or 0.4% ofits measured value. Thus, the maximum error in themeasurement did not exceed 3.0 °C for 760 °C.

The induction heating equipment uses a PID con-troller to regulate the current to obtain a specified tem-perature history. Therefore, a separate measurementthermo-couple was fixed at the workpiece for feedbackto the induction device. Its location was diametricallyopposite to thermo-couple 7 for case A and oppositethermo-couple 4 in test B.

The workpiece material is Inconel 718, and itsproperties can bee seen in Appendix 7. Inconel is para-magnetic and therefore has a permeability of unity.

5 Computational modelA commercial finite element program MSC.

Marc(2005r3 64 bit) was used to solve the computa-tional model. The finite element mesh shown in Fig.3 consist of four-noded axisymmetric quadrilateral el-ement. The different domains, air, coil and workpiece,were given different material properties. Electrical con-ductivity of the coil and air were ignored. The currentwas prescribed in the coil.

The frequency was registered from the heatingequipment and varied for the coil in position A from26 KHz, in the beginning of the experiment, to 22 KHz.A similar behaviour was also registered when the coilwas in position B. See Fig. 4. The average frequencyafter 50s was calculated and was 22.4 KHz for the coilin position A and 23 KHz for the coil located in posi-

tion B. An average frequency of 23 KHz was used in allthe simulations.

Fig. 3. The mesh created for coil position A. Air, coil and workpiece

are modelled.

0 50 100 150 200 250 30021

22

23

24

25

26

27

28

29

Time [s]

Fre

quen

cy [K

Hz]

Coil position ACoil position B

Fig. 4. Measured frequency of the heating equipment.

The heat generation is concentrated near the surfacedue to the skin effect and can be estimated to 3.6 mmwith [9]

δ =1√

π f µσ(18)

Therefore, a fine mesh was created near the surface ofthe workpiece and gradually coarsened towards the in-terior.

Both natural convection and radiation boundarycondition was taken into account during the simulation.The film coefficienth = 1.9 and the potentialα = 1.3was used on all boundaries between the solid and theair [1,10]. The curved surface of the cylinder was non-polished, and is therefore assumed to have an emissivityfactor ofεw = 0.5. The ends of the cylinder were on the

other hand polished and an emissivity ofεw = 0.25 isassumed here.

The following parameters were used in the PIDcontroller logic together with a simulation time step of∆t = 0.5s. See chapter 3 for description of the controllerlogic.

K = 6000

TI = 1080

TD = 210

6 ResultsThe two tests, A and B, differ with respect to the

location of the coil. See Fig. 2. Table 1 gives the loca-tion of the different thermo-couples. Thermo-couples5-7 were used in case A and no. 1, 4 and 5 for case B.

6.1 Controlling current for obtaining specifiedtemperature

The PID controller logic was used to calculate thecurrent needed in the model to obtain the desired tem-perature at a node located at the same place as thermo-couple 7 in case A and 4 in case B. The calculated cur-rent is then compared with the measured current fromthe experiment. The current and temperature at positionA is given in Fig. 5 and 6 respectively. For position B,the current is given in Fig. 7 and Fig. 8 shows the tem-perature history.

6.2 Prescribing measured current to modelThe current profile given from the measurement

was used as input current density (i.e.J0(t)) in thiscase. The calculated temperature was then comparedwith measured temperature. See Fig. 9 for the temper-ature history when the coil is located in position A andFig. 10 for the coil located in position B.

The frequency influence on the temperature wasalso simulated. Three different frequencies, 22, 23 and24 KHz was used as excitation frequencies. The tem-perature was calculated using a prescribed current am-plitude. Fig. 9 shows the temperature history for differ-ent frequencies.

In order to study the influence of current on thetemperature, a small change in the current amplitudewas performed; 1% reduction in case B. The experi-ment followed the same procedure as in section 6.2 andthe result is seen in Fig. 10.

0 50 100 150 200 250 300 3500

20

40

60

80

100

120

140

160

180

Cur

rent

Am

plitu

de [A

]

Time [s]

Fig. 5. Symbols denote measured current and the line denote cal-

culated current using a PID-regulator. The current is the peak value

of the alternating current. This is for test case A.

0 100 200 300 400 500 600 7000

100

200

300

400

500

600

700

800

Time [s]

Tem

pera

ture

[o C]

Calculated dataMeasured data7

6

5

Fig. 6. Measured and computed temperature for case A. The num-

bers refer to measurement locations in Tab. 1. The PID regulator

uses the temperature at position 7 as input.

7 ConclusionsThe good agreement between simulations and mea-

surements shows that the model, the computational ap-proach as well as used material properties, is valid andcan be used to study the induction heating process. Thecurrent can be controlled with a PID-regulator or witha measured current history profile. Regulating the cur-rent work particular well aiming to find the process pa-rameter for a specified temperature history. This willbe useful when designing the induction heating processneeded to create a wanted temperature history.

For test case A, the analysis shows significant dif-ference in temperature between the thermal position 6and 7. This confirms that the induction heating process

0 50 100 150 200 250 300 3500

25

50

75

100

125

150

Time[s]

Cur

rent

Am

plitu

de [A

]

Fig. 7. Symbols denote measured current and the line denote cal-

culated current using a PID-regulator. The current is the peak value

of the alternating current. This is for test case B.

0 100 200 300 400 500 600 7000

100

200

300

400

500

600

700

800

Time[s]

Tem

pera

ture

[o C]

Calculated dataMeasured data

5

1

4

Fig. 8. Measured and computed temperature for case B. The num-

bers refer to measurement locations in Tab. 1. The PID regulator

uses the temperature at position 4 as input.

can produce a desired temperature profile at a selectedregion. The temperature drops fast outside the region ofthe coil, showing that local heat treatment is possible.

The sensitivity of the temperature of different pro-cess parameters (current and frequency) was evaluated.When the current amplitude was reduced by 1% of itsoriginal value, the maximum temperature difference isestimated to be 15 °C. The difference in temperaturebetween the highest and lowest used frequencies is ap-proximately 50 °C. Although the skin depth is smallcompared to the workpiece diameter, it still has an in-fluence on the temperature history. This shows that itmay be important to be able to vary the frequency dur-ing a heating cycle in the computational model.

0 100 200 300 400 500 600 7000

100

200

300

400

500

600

700

800

900

Time [s]

Tem

pera

ture

[o C]

Calculated dataMeasured data

23 KHz22 KHz

24 KHz

7

6

5

Fig. 9. Calculated and measured temperatures for case A when ap-

plying measured current to the model. The current amplitude is seen

in Fig. 5. The frequency is changed to 24 KHz and 22 KHz and

compared with the assumed frequency of 23 KHz.

0 100 200 300 400 500 600 7000

100

200

300

400

500

600

700

800

Time [s]

Tem

pera

ture

[o C]

Calculated dataMeasured data

4

1

5

0.99 x curr. amp.

Fig. 10. Calculated and measured temperatures for case B when

applying measured Curran to the model. The current amplitude is

seen in Fig. 7. The current amplitude is reduces by 1% and com-

pared with the original one.

NomenclatureA magnetic vector potentialB magnetic flux densityD displacement currentE electric field densityH magnetic field densityJ current densityJ0 source current density in the induction coilJe induced current in conductorsJ∗e conjugated complex value ofJK proportional gainP power output

T temperatureQ heat source density generated in the material per

unit volume and timeTa ambient temperature in KelvinTd derivative timeTi integral timeTs surface temperature in Kelvinc specific heat capacitye error between the measured and desired temperaturef frequencyh film coefficienti complex representationn normal vector(n) current time step(n−1) previous time stepqn the outgoing heat fluxu controller output∆t time step lengthδ skin depthε permittivityεw emissivity of the workpieceκ the thermal conductivityµ permeabilityω excitation frequencyρ density of the materialρv free volume chargesσ electrical conductivityσs Stefan Boltzmann’s constant

References[1] Rudnev, V., Loveless, D., Cook, R., and Black,

M., 2003. Handbook of Induction Heating. In-ductoheat, Inc.

[2] Clain, S., Rappaz, J., Swierkosz, J., and Touzani,R., 1993. “Numerical modeling of induction heat-ing for two-dimentional geometries”.Mathemat-icl Models and Methods in Applied Sciences,3(6),p. 805–822.

[3] Wang, K., Chandrasekar, S., and Yang, H. T.,1992. “Finite-element simulation of inductionheat treatment”.Journal of Materials Engineer-ing and Performance,1.

[4] Chaboudez, C., Clain, S., Glardon, R., Mari, D.,Rappaz, J., and Swierkosz, M., 1997. “Numeri-cal modeling in induction heating for axisymmet-ric geometries”. Magnetics, IEEE Transactionson,33(1), p. 739–745. 0018-9464.

[5] Luo, J., and Shih, A. J., 2005. “Invers heat transfersolution of the heat flux due to induction heating”.

Journal of Manufacturing Science and Engineer-ing, 127, August, p. 555–563.

[6] Bay, F., Labbe, V., Favennec, Y., and Chenot, J. L.,2003. “A numerical model for induction heatingprocesses coupling electromagnetism and thermo-mechanics”. International Journal for Numeri-cal Methods in Engineering,58(6), p. 839–867.10.1002/nme.796.

[7] Drobenko, B., Hachjevych, O., and Kournyts’kyi,T., 2006. “A mathematical simulation of hightemperature induction heating of electrocunduc-tive solids”. International Heat and Mass Trans-fer.

[8] Stratton, J. A., 2007. Electromagnetic theory.IEEE Press, Piscataway, NJ.

[9] Sadiku, M. N., 2001.Elements of Electromagnet-ics, Vol. 3. Oxford University Press, Inc.

[10] Davies, E. J., 1990.Conduction and inductionheating. No. 0-86341-174-6 in IEE Power Enge-neering Series 11. Peter Peregrinus Ltd., London,UK.

APPENDIXThe used material properties for Inconel 718. Fig.

11 shows the electric and thermal conductivity and Fig.12 the specific heat capacity. The electric conductivityis expressed in Kilo Siemens per meter.

0 200 400 600 800 1000 120010

12

14

16

18

20

22

24

26

28

30

Temp [°C]

The

rmal

con

duct

ivity

[W/m

K]

Ele

ctric

con

duct

ivity

[KS

/m]

750

760

770

780

790

800

810

820

830

840

Fig. 11. The electric and thermal conductivity for Inconel 718.

0 200 400 600 800 1000 1200400

450

500

550

600

650

700

Temp [°C]

Spe

cific

Hea

t Cap

acity

[J/K

g K

]

Fig. 12. The heat capacity in constant pressure for Inconel 718.

Paper B

Simulations and measurements of ombined indu tion heating and extrusion pro essesS. Hanssona,b,∗, M. Fisk aSandvik Materials Te hnology, 811 81 Sandviken, SwedenbDalarna University, 781 77 Borlänge, Sweden Luleå University of Te hnology, Division of Material Me hani s, 971 87 Luleå, SwedenAbstra tThe manufa turing pro ess hain at glass-lubri ated extrusion of stainless steel tubes is simulated using the nite elementmethod. The developed model in ludes sub-models of indu tion heating, expansion and extrusion. An in-house mappingtool is used to transfer the temperature elds between the ele tromagneti -thermal and thermo-me hani al analyses.The model is su essfully applied to two ases of tube extrusion; one using an austeniti stainless steel and one usinga duplex, austeniti /ferriti stainless steel. It is shown that the indu tion heating model predi ts the temperaturesobtained experimentally from thermo ouples pla ed in the steel billets during heating. The agreement between modelsand experiments regarding extrusion for e and expansion for e are satisfa tory.Key words: Eddy urrent, Stainless steel, Finite Element Method, Extrusion, Ferromagneti , Magneti hysteresis.1. Introdu tionSeamless stainless steel tubes an be manufa tured byhot extrusion using glass as lubri ation. Before extrusionthe billet is prepared in several operations, in luding in-du tion heating, lubri ation and expansion.Simulation of extrusion using the nite element method(FEM) has been performed extensively during the lastde ade. The majority of these simulations on ern extru-sion of aluminum but work regarding steel and titaniumextrusion has also been reported. The most ommon ap-proa h in these simulations is to use a uniform billet tem-perature as initial ondition in the extrusion model. Theinitial stages of heating and transport are often ignored.However, the major ee ts of extrusion variables on pres-sure in steel extrusion are related to their ee t on thetemperature of the billet, and hen e the ow stress of thematerial [3. Knowledge of the billet temperature prolebefore extrusion is therefore signi ant to obtain a uratesimulation results. It is also important to understand theheating pro ess in order to eliminate unwanted tempera-ture gradients.In this study, axisymmetri FE-models of indu tionheating, expansion and extrusion, in luding ooling at theintermediate transports, have been ombined. The pro- ess ow was modeled using seven dierent sub-models, asshown in Fig. 1. All models were developed in the om-mer ial FE-software MSC.Mar and the nodal tempera-tures were transferred from one model to another usingan external mapping program. No strains were mapped∗Corresponding authorEmail addresses: sofia.e.hanssonsandvik. om (S. Hansson),martin.fiskltu.se (M. Fisk)

from previous sub-model, i.e. the material is assumed tobe fully re overed at the initial stage of ea h model.Extrusion of two dierent stainless steel tubes was stud-ied. One was made of an austeniti grade, Sandvik 353MA, and one of a duplex, austeniti /ferriti alloy, SandvikSAF 2507. Sin e the duplex steel is ferromagneti belowthe Curie temperature, the skin ee t is more pronoun ed ompared to the paramagneti austeniti material. Alsothe magneti non-linearity of the material properties leadsto dieren es in the indu tion heating modelling. Resultsfrom the numeri al models have been validated by exper-iments in produ tion presses and indu tion furna es.Figure 1: Simulated pro ess ow. Ea h box refers to one sub-model.2. Pro ess owA s hemati diagram of the pro ess ow an be seen inFig. 1. The material is re eived as a hot rolled round barwhi h is turned and ut into lengths based on the materialvolume required for the extruded tube. A pilot bore holeis drilled in ea h billet; a nose radius is ma hined at thefront edge and a one at the ba k end. After this, the billetis transported for manufa turing, where the rst step is toheat the billet in an indu tion furna e. After heating, thebillet is rolled down a glass- overed tray until the surfa eis oated with a thin layer of powdered glass. This glassis used as lubri ant in the following expansion pro ess.Preprint submitted to Elsevier November 13, 2008

The heated billet is inserted nose down in a verti alexpansion press, in whi h the pilot bore hole is expandedusing a shaped nose. The billet is elongated during thispro ess. A glass pad (a dis of ompa ted glass) is pla edin front of the nose and melts during the operation, therebyassuring su ient lubri ation. In the expansion pro ess, itis desirable to have a ontrolled front temperature of thebillet. This is to avoid unwanted extrusion at the bottomof the press.The expanded billet is reheated, lubri ated and ex-truded to the spe ied nal dimension. A glass pad ispla ed between the billet and the die to ensure lubri a-tion. In this operation, a hot leading end is desired inorder to redu e the peak extrusion for e.The billet is heated in an indu tion furna e with theaim to provide a uniform surfa e-to- ore temperature. How-ever, in indu tion heating the ore is heated slower than itssurfa e. This is due to the skin ee t. The skin depth de-pends on the frequen y of the indu tion heating power aswell as the material properties of the billet [5. It should benoti ed that it is di ult to a hieve uniform temperaturein stainless steel due to its poor thermal ondu tivity.The basi setup of an indu tion furna e onsists of a oil with one or several windings, keeping an ele tri ally ondu ting workpie e within. The urrent arrying oilprodu es an alternating ele tromagneti eld whi h in-du es eddy urrent in the surfa e of the workpie e. Heatis generated due to the Joule heating ee t aused by this urrent.

Figure 2: The pro ess ow for produ tion of stainless steel tubes.3. The ele tromagneti -thermal problemSimpli ations employed in the al ulation of the heate ien y fa tor in indu tion heating and linearisation ofthe ferromagneti material are summarised below.3.1. Cal ulation of the heat e ien y fa torThe magneti eld of the oil in the indu tion furna eindu es voltage in the workpie e as well as in the oil itself.The indu ed voltage is in su h dire tion that it tends toweaken the urrent owing in the oil. In order to simulatethis phenomena using the nite element method, a double

onne ted ondu tor formulation must be employed [1.This is not implemented in MSC.Mar and this ee t isapproximated as des ribed below.The resistan e in an indu tion heating system is thesum of the resistan e of the oil and the ree ted resistan eof the workpie e [8. It is thus possible to represent the oil and workpie e resistan e as a series ir uit.This allows us to al ulate a heat e ien y fa torη =

Rw

Rw + Rc

(1)where Rc is the resistan e of the oil and Rw, is the re-sistan e of the workpie e. Details an be found in [10, 2.The heat e ien y fa tor is used to redu e the heat sour eterm in the heat ondu tion equation.The resistan e of a ylindri al workpie e and its oil an be written as [2,Rw = K(µrpAw) (2)Rc = K(

krπdcδc

2) (3)Here is K a onstant, µr is the relative permeability, pis the power loss onstant, Aw is the ross se tion area ofthe workpie e, kr is the orre tion fa tor allowing spa ingbetween oil turns, dc is the oil diameter and δc is theskin depth of the oil. The skin depth of a material anbe al ulated as [6

δ =1√

πfµσ(4)where f is the ex itation frequen y, µ the permeabilityand σ the ele tri al ondu tivity of the material.The e ien y fa tor, η, an now be al ulated for aspe i oil and workpie e geometry and used to multiplythe heat sour e term in the heat ondu tion equation

ρcp

∂T

∂t−∇ · (κ∇T ) = ηQ (5)where ρ is the density of the material, cp is the spe- i heat apa ity and κ is the thermal ondu tivity ofthe workpie e. In the ase of a harmoni ele tromagneti loading the heat sour e term due to the eddy urrent anbe written as [1, 9

Q =1

2σ|J|2 =

1

2σJ · J∗ (6)where J is the indu ed eddy urrent in the workpie e andthe star (*) denotes the onjugated omplex value of the urrent J. The urrent is genereted within the workpie eas J = −iωσA where A is the magneti ve tor potential al ulated from the diusion equation. See Eq. (7) inse tion 3.2. All material properties are fun tions of tem-perature.2

3.2. Magneti o-energy density and non-linearityIn the nite element implementation, time variationof the eld quantities are assumed to be sinusoidal. Tobe able to use this harmoni approximation, the magneti material property must be onstant. Combining Maxwell'sequations, the linear diusion equation an be formulated[7 asiωσA− 1

µ∇2A = J0 (7)where A is the magneti ve tor potential, ω is the angularfrequen y of the magneti eld, µ is the permeability ofthe material, and J0 is the urrent applied in the oil.The ferromagneti material SAF 2507 has nonlinearmagneti properties and therefore Eq. (7) is not valid. Thehysteresis urve for SAF 2507 at dierent temperatures an be seen in Fig. 2. The magneti ux density, B,represents the internal eld strength within a substan e,whi h is subje ted to an external magneti eld, H. Thishysteresis behavior an be modeled by the Fröli h equationB =H

α + β||H|| (8)where α and β are variables dependent on temperature.Figure 3 shows the values of α and β for SAF 2507.The nonlinear ele tromagneti behavior of the material an be linearised following Labridis and Dokopoulos [4.They have shown that it is possible to approximate thenonlinear material with a tive linear material that has onstant relative permeability, µfr . The linear titiousmaterial should have the same average heat density loss asthat of the nonlinear material at every point i. It an beshown that the tive relative permeability is [4

µfri =

w1i + w2i

µ0 ∗ (Hfmi)

2(9)where w1i and w2i is the magneti o-energy density of thematerial. w1i is related to the a tual B−H urve and w2iis related to the average value of the slope dB/dH duringa quarter of a period T . Hf

mi is the maximum value ofthe magneti eld intensity in the titious material. Themagneti o-energy density an be al ulated asw1i =

Hfmi

β− α

β2ln

α + βHfmi

α(10)

w2i =1

2

(Hfmi)

2

α + βHfmi

(11)Sin e both the relative permeability of the titiousmaterial and the magneti eld intensity are unknown, aniterative pro ess is ne essary. The starting value of µfr isset to the initial slope of the Fröli h urve, i.e.

µfr =

1

µ0α(12)Thereafter µf

r is al ulated from Eq. (9).

−5 −4 −3 −2 −1 0 1 2 3 4 5

x 105

−1.5

−1

−0.5

0

0.5

1

1.5

H [A/m]

B [V

s/m

2 ]

20100200300400450480490500520600Figure 3: Magneti hysteresis urve for dierent temperatures inSAF 2507. The Curie temperature starts approximately at 500 C.

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Temperature [°C]

β α

0

1

2

3

4

5

6

7

8x 10

5

Figure 4: The variation of α and β for dierent temperatures. SeeEq. (8).4. MaterialsTwo dierent grades of stainless steels, Sandvik 353MA and Sandvik SAF 2507, were onsidered in this study.353 MA (UNS: S35315) is an austeniti hromium-ni kelsteel mainly used in high temperature appli ations. SAF2507 (UNS: S32750) is a high alloyed duplex grade, onsist-ing of approximately 50% austenite in a matrix of ferrite.The nominal hemi al ompositions of 353 MA and SAF2507 are given in Table 1 and Table 2, respe tively.4.1. Flow stressThe deformation behavior at high temperature was de-termined by me hani al testing in a number of ompres-sion tests. These tests were performed in a Gleeble 1500thermome hani al simulator at several temperatures and3

Table 1: Chemi al omposition (nominal) of 353 MA [%C Si Mn P S Cr Ni N Cemax max0.07 1.6 1.5 0.040 0.015 25 35 0.16 0.05Table 2: Chemi al omposition (nominal) of SAF 2507 [%C Si Mn P S Cr Ni Mo Nmax max max max max0.030 0.8 1.2 0.035 0.015 25 7 4 0.3strain rates, up to a total strain of 60%. Cylindri al spe i-mens of diameter 10mm and height 12mm were ma hinedfrom hot rolled round bars, with the rolling dire tion ofthe bar orresponding to the loading axis of the test spe -imens.Flow stress urves of SAF 2507 at 1000, 1100, 1200and 1300 C and strain rates 0.01, 1 and 10 s−1 are givenin Fig. 4 and Fig. 5. The jerky ow and the serrationsthat are visible at higher deformation rates are dynami ee ts due to the testing ma hine.No me hani al testing was ondu ted for 353 MA. Itsmaterial properties were assumed to be equal to those ofSandvik Sani ro 28, a similar material. A omparison ofthe hemi al ompositions of the materials an be seen inTable 1. Flow stress urves of Sani ro 28 at 1000, 1100,1200 and 1300 C and strain rates 0.01, 1 and 10 s−1 aregiven in Fig. 6 and Fig. 7.The similarity in hot strength of Sani ro 28 and 353MA are illustrated in Fig. 8. These tests were arried outin tension at a onstant rate of 50mm/s. The test spe -imens were ylindri al rods, length 50 mm and diameter6.3 mm, whi h were strained to fra ture at dierent tem-peratures. The maximum for e was re orded and used asan estimation of the hot du tility in the material.5. Experiments5.1. Indu tion furna eVerti al indu tion furna es with water ooled oil sur-rounding the billet are used in the experiments. The billetis inserted into the oil using a hydrauli ram. Flux on- entrators are applied at both ends of the billet in orderto obtain a required temperature uniformity through thelength of the billet. Using ux on entrators, the oil over-hang an be minimized and the possible billet length in agiven furna e an be maximized [5. The number of oilturns an be varied to obtain a desired temperature prole.The billet is heated by applying an alternating urrentto the oil. When the surfa e temperature rea hes a spe -ied value (usually between 1170 − 1210 C), the furna eis swit hed o, letting the billet ool down inside the fur-na e. After a number of se onds, the furna e is turned

0 0.1 0.2 0.3 0.4 0.5 0.60

50

100

150

200

250

300

350

True Strain [−]

Tru

e S

tres

s [M

Pa]

1000°C, 10/s

1000°C, 1/s

1000°C, 0.01/s

1100°C, 10/s

1100°C, 1/s

1100°C, 0.01/s

Figure 5: Compression tests of SAF 2507 at temperatures 1000 and1100 C.

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

True Strain [−]

Tru

e S

tres

s [M

Pa]

1200°C, 10/s

1200°C, 1/s

1200°C, 0.01/s

1300°C, 10/s

1300°C, 1/s

1300°C, 0.01/sFigure 6: Compression tests of SAF 2507 at temperatures 1200 and1300 C.on again for a nal heating before the billet is eje ted andtransported for lubri ation and expansion. To y le thepower leads to more uniform surfa e-to- ore temperature,whi h is desirable in the extrusion pro ess. The same typeof furna e is used for the heating between expansion andextrusion (Heating 2), although dierent number of a tive oil and no power y ling are used. See Fig. 1 for thesimulation steps.The oil liner has a thi kness of 19mm with an innerdiameter of 360mm. The furna e onsists of refra torymaterial and a metal shell to prote t the oil liner fromdamage and redu e the heat loss to the environment. Thisredu es the inner diameter of the furna e to 260mm. Thefrequen y of the urrent is 50Hz.The temperature of the billet during heating was mea-sured ea h se ond with K-type thermo ouples using a PC-4

0 0.1 0.2 0.3 0.4 0.5 0.60

50

100

150

200

250

300

350

400

True Strain [−]

Tru

e S

tres

s [M

Pa]

1000°C, 10/s

1000°C, 1/s

1000°C, 0.01/s

1100°C, 10/s

1100°C, 1/s

1100°C, 0.01/s

Figure 7: Compression tests of Sani ro 28 at temperatures 1000 and1100 C.

0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

160

180

200

220

True Strain [−]

Tru

e S

tres

s [M

Pa]

1200°C, 10/s

1200°C, 1/s

1200°C, 0.01/s

1300°C, 10/s

1300°C, 1/s

1300°C, 0.01/s

Figure 8: Compression tests of Sani ro 28 at temperatures 1200 and1300 C.logger. The thermo ouples were pla ed in drilled holes atdierent heights and depths of the billet. See Fig. 9 andTable 3 for the pla ement of the thermo ouples. The oil urrent was re orded ea h se ond using a PC-logger and an be seen in Fig. 10.5.2. Expansion and extrusionThe expansion and extrusion pro esses are ondu tedin a 12MN verti al hydrauli press and a 33MN horizontalhydrauli press, respe tively. All experimental data wasre orded during produ tion in these presses. In the expan-sion press, the ontainer diameter was 240mm with a max-imum nose diameter of 112mm for 353 MA and 118mmfor SAF 2507. The transient for e during expansion wasre orded for SAF 2507 and the peak for es for 353 MA.The ontainer diameter in the extrusion press was250mm. The mandrel and die geometry was hosen in

850 900 950 1000 1050 1100 1150 1200 1250 13001000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

Temperature [oC]

Max

For

ce [N

]

Sandvik 353 MASandvik Sanicro 28

Figure 9: Comparison of hot du tility in Sani ro 28 and 353 MA.Figure 10: Pla ement of thermo- ouples (x) and geometri propertiesof the billet.a ordan e with the dimensions of the nal tube. Thesemeasurements are given in Tables 4 and 5. Transient for eand ram speed were re orded during the extrusion pro ess.6. Computational modelsSeparate models for expansion, extrusion and indu -tion heating pro esses are used. The temperatures of thepre eding pro ess step is mapped with an in-house odeto the initial mesh of the model in the following step. The ommer ial nite element program MSC. Mar 2007r1 64bit is used for all al ulations.Table 3: Pla ement of the thermo ouples.Steel Billet dimension a btype LxRxr [mm [mm [mm353 MA 738x118.5x18.5 100 10, 20, 72369 10, 20, 94638 10, 50, 94SAF 2507 946x118.5x22.5 473 5, 10, 20, 30,40, 64, 785

0 100 200 300 400 500 600 700 800 9000

500

1000

1500

2000

2500

3000

3500

4000

4500

Time [s]

Cur

rent

Am

plitu

de [A

]

SAF 2507 − Heating 2

535 MA − Heating 2

535 MA − Heating 1

SAF 2507 − Heating 1

Figure 11: Nominal urrent amplitude in the indu tion oil duringthe heating pro edures.Table 4: Billet and tube dimensions used in 353 MA simulations.Billet Outer diameter [mm 237Inner diameter [mm 37Length [mm 738Expanded billet Outer diameter [mm 240Inner diameter [mm 112Extruded tube Outer diameter [mm 141Wall thi kness [mm 22.0Table 5: Billet and tube dimensions used in SAF 2507 simulations.Billet Outer diameter [mm 237Inner diameter [mm 45Length [mm 946Expanded billet Outer diameter [mm 240Inner diameter [mm 118Extruded tube Outer diameter [mm 114Wall thi kness [mm 10.06.1. Indu tion heatingThe FE-models are meshed with four-noded axisym-metri quadrilateral elements. See Fig. 11 for how themesh is reated for Heating 1 of 353 MA. The radiationheat ux is ignored as the refra tory material inside thefurna e is assumed to ree t the heat radiated from thebillet.The heat transfer by onve tion from the surfa e of theworkpie e to the ambient air is taken into a ount. A goodapproximation for a workpie e in open air [5, 2 is,qconv = 1.54(Tx − Ta)1.33 (13)where Ta is the ambient temperature. Sin e the ambienttemperature inside the furna e is hanging during heating,

this temperature is a fun tion of time. The oil liners havea onstant temperature of 60 C.In order to simulate the ee t of the ux on entrators,the material properties from SAF 2507 are used. The non-linear ele tromagneti behavior of the ux on entratormaterial is also linearised. The permeability is, however,multiplied with a fa tor of twenty, sin e the ux on en-trators must have a higher permeability than SAF 2507[5.6.1.1. Sandvik 353 MAThe workpie e resistan e and the oil resistan e are al ulated using Eq. (2) and Eq. (3). With numbers theygiveRw = K(µrpAw) = K(1 · 0.35 · 0.043) =

= K0.015 (14)Rc = K

(

krπdcδc

2

)

=

= K

(

1.05 · π · 0.292 · 0.009

2

)

=

= K0.0045 (15)The heat e ien y fa tor an then be al ulated from Eq.(1), giving η ≈ 0.78.The oil overhang (the distan e from the rst a tive oilturn to the front end of the workpie e) during Heating 1and Heating 2 are 115 mm and 185 mm. Number of a tive oil turns are 48 and 58, respe tively. The nominal urrentin the oil is seen in Fig. 10. In Heating 2, the e ien yfa tor is al ulated to be η = 0.74.Figure 12: FE-model of the indu tion furna e for Heating 1 of 353MA. The smallest element size is 2 mm.6.1.2. SAF 2507In a ferromagneti material, the initial skin depth issmaller than in a paramagneti material, resulting in ahigher e ien y fa tor. Above the Curie temperature, thee ien y fa tor will hange to that of a paramagneti ma-terial, be ause of de reasing magnetism for in reasing tem-peratures. However, a onstant e ien y fa tor (η = 0.85)is used throughout the Heating 1 simulation.The oil overhang in Heating 1 is 95 mm and in Heating2 155 mm. Number of a tive oil turns are 59 and 72. Thenominal urrent in the oil is seen in Fig. 10. In Heating2 the e ien y fa tor was al ulated to be η = 0.73.6

6.2. CoolingBoth radiation and onve tion heat losses are takeninto a ount in the ooling stages. The ambient tempera-ture is xed to Ta = 303K. The onve tion losses are thesame as in se tion 6.1 and the surfa e emissivity is ε = 0.3.The total ux an therefore be expressed as where σs isthe Stefan Boltzmann's onstant.The transport time from the indu tion furna e to theexpansion press (Cooling 1) is 70 s. The expanded billetis transported 60 s to the nal heating (Cooling 2), andthereafter 70 s to the extrusion press (Cooling 3). See Fig.1 for the simulation pro ess ow.6.3. Expansion and extrusionThe expansion and extrusion models are built up usingfour-node axisymmetri quadrilateral elements. The toolsare treated as rigid bodies with heat transfer properties.This means that the heat transfer between the billet andthe tools is aptured but the dee tion of the tools is ig-nored. The isolating properties of the lubri ating glass lmimplies low onta t heat transfer relative to other lubri- ants. A onta t heat transfer oe ient of 1.5 kW/m2Kis used at the glass lubri ated onta t areas. The toolsin the models have thermal properties equal to the ma-terial AISI H13 and an initial temperature of 400 C atexpansion and 300 C at extrusion. The Coulomb fri tion oe ient between the billet and the tools is assumed toµC = 0.02 at all glass lubri ated onta t areas. In extru-sion, the fri tion oe ient between the billet and the ramis µC = 0.35.The ram (nose) speed in the expansion pro ess is160mm/s for 353 MA and 200mm/s for SAF 2507. Inthe extrusion pro ess, the ram speed is 260mm/s for 353MA and 210mm/s for SAF 2507.Generation of heat due to plasti deformation in thematerial is taken into a ount and 95% of the plasti workis assumed to be onverted into thermal energy. Large de-formations in the Lagrangian ode implies that ontinuousremeshing is needed during the analysis. This is ontrolledby automati remeshing s hemes available in the software.The rst and last in rement of the 353 MA expansionmodel an be seen in Fig. 12 and Fig. 13, respe tively. TheFE mesh for extrusion of the same material is shown in Fig.14. The models for SAF 2507 expansion and extrusion areidenti al to those with 353 MA apart from dieren es ingeometry (see Tables 4 and 5), ram speed and materialmodel.

Figure 13: FE-model of expansion of 353 MA (initial stage).

Figure 14: FE-model of expansion of 353 MA (nal stage).Figure 15: FE-model of extrusion of 353 MA (initial stage).6.3.1. Material modelThe materials are modeled as elasti -plasti with iso-tropi hardening using the von Mises yield riterion. Theow stress dependen e on temperature, plasti strain andstrain rate is tabulated. The underlying experimental data,see Figs. 4-7, in ludes temperatures between 800−1300 Cand strain rates between 0.01 − 10 s−1. Linear interpola-tion is used inside this domain. Outside the temperature,strain, and strain rate limits, the extreme values are usedand no extrapolation is performed.6.4. Mapping between indu tion heating and extrusionAn in-house mapping tool for moving data between twonite element models is used to map temperature elds be-tween the ele tromagneti -thermal and thermo-me hani alanalysis. First a material identi ation is performed. Ea hmaterial of the re eiving model obtains temperatures onlyfrom orresponding material of the sending model. Mate-rials in the re eiving model whi h have no orrespondingmaterial in the sending model, are given a user denedtemperature.The mapping method is based on the interpolationte hnique used in nite element formulations. The nodaltemperatures of the re eiving model are obtained by

Tnew = Nold(Sloc)Told (16)where Nold is the shape fun tion of the element in thesending model evaluated at the lo al oordinate, Sloc. Inthis pro edure it is required to determine where a nodeof the re eiving mesh is lo ated in the sending mesh. Asear h is performed to determine in whi h element of thesending (old) mesh the node is lo ated, and whi h lo al oordinate it orresponds to. The use of isoparametri nite elements givesXnew = Nold(Sloc)Xold (17)where Xold is the nodal oordinates of the sending elementand Xnew is the oordinate of the node whi h re eives thetemperature.Eq. (17) denes a one-to-one mapping between the lo- al oordinate system and the global oordinate system of7

the mesh. However, in the general ase, it does not existan analyti al inverse mapping in whi h the lo al oordi-nate, Sloc, an be written expli itly. Thus Sloc must beevaluated numeri ally by solving||Xnew − Nold(Sloc)Xold|| = 0 (18)This is done by the Newton iterative method where theinitial value for the lo al oordinate is guessed.7. Results and dis ussionThe measured and omputed values of temperaturesand for es are plotted in the Figures below. Fig. 15 - 17show the temperatures for 353 MA and Fig. 18 to 19 showthe temperatures for SAF 2507. All the temperatures aremeasured during manufa turing step Heating 1 with thethermo ouples pla ed a ording to Fig. 9.The expansion for e versus time for 353 MA is shown inFig. 20. The extrusion for e versus the ram displa ementfor the same material is shown in Fig. 21. Figure 22 andFig. 23 give the expansion for e versus time and extrusionfor e versus ram displa ement for SAF 2507.7.1. Indu tion heatingThe results from the indu tion heating pro ess showgood agreement between measured and omputed values.For the material 353 MA the temperature is measured atthree dierent height and at three dierent depths as de-s ribed in se tion 5.1. At the depth of 50 − 72mm in theworkpie e, the measured and omputed values show max-imum deviation.

0 100 200 300 400 500 600 7000

200

400

600

800

1000

1200

1400

Time [s]

Tem

pera

ture

[°C

]

10 mm20 mm72 mm

Figure 16: The measured temperatures (dashed lines) ompared with al ulated temperatures (full lines) at position a = 100 mm in 353MA.For the material SAF 2507 the temperature is mea-sured in the middle of the workpie e at six dierent depths.The temperature agreement for the duplex steel shows that

0 100 200 300 400 500 600 7000

200

400

600

800

1000

1200

1400

Time [s]

Tem

pera

ture

[°C

]

10 mm20 mm94 mm

Figure 17: The measured temperatures (dashed lines) ompared with al ulated temperatures (full lines) at position a = 369 mm in 353MA.

0 100 200 300 400 500 600 7000

200

400

600

800

1000

1200

1400

Time [s]

Tem

pera

ture

[°C

]

10 mm50 mm94 mm

Figure 18: The measured temperatures (dashed lines) ompared with al ulated temperatures (full lines) at position a = 638 mm in 353MA.the approa h of Labridis and Dokopoulos [4 to linearizethe permeability is valid. This approximation makes itpossible to employ a harmoni formulation for non-linearmagneti material. This is important sin e a harmoni so-lution with a onstant permeability is mu h faster than atransient ele tromagneti non-linear solution [1.In Fig. 18 it an be seen that the heating rate is greaterwhen the material is ferromagneti . When the materialbe omes paramagneti , the heating rate de reases. This an be seen as a knee in the temperature graph that startsat dierent times for dierent depths, depending on whenthe Curie temperature is rea hed.Figure 19 shows the last 140 s of the heating simulationof SAF 2507. From the gure it an be seen that the mea-sured and al ulated temperatures losest to the surfa e8

(10mm) are parallel during the power y ling. This showsthat the heat ux boundary ondition is valid.

0 100 200 300 400 500 600 700 800 9000

200

400

600

800

1000

1200

1400

Time [s]

Tem

pera

ture

[°C

]

10 mm20 mm30 mm40 mm64 mm78 mm

Figure 19: The measured temperatures (dashed lines) ompared with al ulated temperatures (full lines) at position a = 473 mm in SAF2507.

660 680 700 720 740 760 780 8001050

1100

1150

1200

1250

1300

Time [s]

Tem

pera

ture

[°C

]

10 mm20 mm30 mm40 mm64 mm78 mm

Figure 20: The measured temperatures (dashed lines) ompared with al ulated temperatures (full lines) at position a = 473 mm in SAF2507.7.2. Expansion and extrusionExperimental values were re orded for 14 tubes of 353MA. Only the peak for es ould be re orded during ex-pansion and were measured to be 4.2 ±0.5MN and 4.6±0.4MN. This is in agreement with the model predi -tions as shown in Fig. 20, although the se ond peak is onthe lower side.The 353 MA billets in the models were shorter than theones on whi h the for e measurements were arried out.The experimental for e re ordings were performed on bil-lets with lengths of 810mm before expansion and 960mm

before extrusion, ompared to 738mm and 890mm in thesimulations. The dieren e in length an be seen in Fig.21. The temperature measurement during Heating 1 isperformed on the shorter billet.

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time [s]

Exp

ansi

on F

orce

[MN

]

Figure 21: Cal ulated and measured for e for extrusion of 353 MA.

0 200 400 600 800 10000

5

10

15

20

25

30

35

Ram Displacement [mm]

Ext

rusi

on F

orce

[MN

]

Figure 22: Cal ulated (bla k line) and measured for e (blue line) forextrusion of 353 MA.The predi tion of the initial peak for e during extru-sion is higher in the models than in the experiments. Thisis probably due to errors in the temperature estimation atthe leading end. The ow stress of 353 MA has pronoun edtemperature dependen e and a small hange in tempera-ture will result in a large dieren e in stress. However, ifthe 353 MA extrusion simulation is run using a homoge-neous initial billet temperature of 1200 C, the resultingpeak for e is only 1.5MN higher than the ow for e atsteady-state. In the experiments the dieren e betweeninitial peak and ow for e is approximately 9MN as seenin Fig. 21. This laries that a orre t front fa e temper-9

ature of the billet is important in determining the initialpeak.Figure 21 and Fig. 23 reveal that the start-up phaseof extrusion is not fully aptured in the model. The mainreason is the un ertainties of the onditions at the begin-ning of the extrusion pro ess. The glass-pad deforms in onta t with the metal, making it di ult to predi t theinterfa e between the billet, the glass pad and the die. Inthe model this is negle ted and a onstant die shape anda onstant fri tion oe ient is used. Furthermore, grav-ity is negle ted in the axisymmetri model. In the realprodu tion ase, the billet is pla ed at the bottom of thehorizontal press and gravity will inuen e how the metallls the ontainer.

0 1 2 3 4 50

0.5

1

1.5

2

2.5

3

Time [s]

Exp

ansi

on F

orce

[MN

]

Figure 23: Cal ulated (bla k line) and measured for e (blue line) forexpansion of SAF 2507

0 200 400 600 800 1000 12000

5

10

15

20

25

30

35

Ram Displacement [mm]

Ext

rusi

on F

orce

[MN

]

Figure 24: Cal ulated (bla k line) and measured for e (blue line) forextrusion of SAF 2507

8. Con lusionsA ombined model of indu tion heating and expan-sion/extrusion, onsisting of seven dierent FE-models,was developed in order to study the evolution of tempera-ture gradients in stainless steel billets for tube manufa tur-ing. Simulations were arried out for an austeniti stain-less steel, Sandvik 353 MA, and for a duplex (austeniti /ferriti ) stainless steel, Sandvik SAF 2507. The indu -tion heating models were validated by heating experimentsusing thermo ouples pla ed in holes drilled at dierentpositions in the billet. The agreement between modeland experiments was good, both for the paramagneti austeniti material and the ferromagneti duplex material.The agreement for SAF 2507 onrms that the approa h tolinearize the permeability is valid. The expansion and ex-trusion models were able to apture the for e developmentduring the pro esses for both materials. When omparingto experimental re ordings of expansion for e versus time,and extrusion for e versus ram displa ement, the resultswere satisfa tory.A knowledgementsThe authors are grateful to prof. Lars-Erik Lindgrenat the Division of Material Me hani s at Luleå Univer-sity of Te hnology, Luleå, Sweden for programming themapping algorithm and to Christofer Hedvall and BerndtFerm at Sandvik Materials Te hnology, Sandviken, Swe-den for their assistan e in the experimental work. The -nan ial support from the Swedish Knowledge Foundationand the Swedish Steel Produ ers' Asso iation is gratefullya knowledged.Referen es[1 Clain, S., Rappaz, J., Swierkosz, J., Touzani, R., 1993. Numer-i al modeling of indu tion heating for two-dimensional geome-tries. Mathemati al Models and Methods in Applied S ien es3 (6), 805822.[2 Davies, E. J., 1990. Condu tion and indu tion heating. No. 0-86341-174-6 in IEE Power Engeneering Series 11. Peter Peregri-nus Ltd., London, UK.[3 Hughes, K., Nair, K., Sellars, C., 1974. Temperature and owstress during the hot extrusion of steel. Metals Te hnology 1 (4),161169.[4 Labridis, D., Dokopoulos, P., May 1989. Cal ulation of eddy urrent losses in nonlinear ferromagneit materials. IEEE Trans-a tions on Magneti s 25 (3).[5 Rudnev, V., Loveless, D., Cook, R., Bla k, M., 2003. Handbookof Indu tion Heating. Indu toheat, In .[6 Sadiku, M. N., 2001. Elements of Ele tromagneti s. Vol. 3. Ox-ford University Press, In .[7 Stratton, J. A., 2007. Ele tromagneti theory. IEEE Press, Pis- ataway, NJ.[8 Tudbury, C. A., 1960. Basi s of Indu tion Heating. Vol. 2. JohnF. Rider, In .[9 Wang, K., Chandrasekar, S., Yang, H. T., 1992. Finite-elementsimulation of indu tion heat treatment. Journal of MaterialsEngineering and Performan e 1.[10 Zinn, S., Semiatin, S. L., 1988. Elements of Indu tion Heating.Ele tri Power Resear h Institute, In .10

A. Data for Sandvik 353 MA and Sandvik SAF2507

0 200 400 600 800 1000 1200 14000.7

0.8

0.9

1

1.1

1.2

1.3x 10

6

Temperature [oC]

Ele

ctric

Con

duct

ivity

[S/m

]

353 MASAF 2507

Figure 25: The ele tri ondu tivity for Sani ro 39 and SAF 2507.

0 200 400 600 800 1000 1200 140010

15

20

25

30

35

Temperature [oC]

The

rmal

Con

duct

ivity

[W/m

K]

353 MASAF 2507

Figure 26: The thermal ondu tivity for Sani ro 39 and SAF 2507.0 200 400 600 800 1000 1200 1400

450

500

550

600

650

700

750

800

850

900

950

Temperature [oC]

Spe

cific

Hea

t Cap

acity

[J/K

g K

]

353 MASAF 2507

Figure 27: The heat apa ity in onstant pressure for Sani ro 39 andSAF 2507.

11