Scientific Research and Essays Vol. 8(2), pp. 48 -61, 11 January, 2013 Available online at http://www.academicjournals.org/SRE DOI: 10.5897/SRE11.1880 ISSN 1992-2248 ©2013 Academic Journals

Full Length Research Paper

Influence of cross section shape of the coil turns on the induction heating process

Mohammmad Hossein Tavakoli*, Mohtaram Honarmandnia and Hamed Heydari

Physics Department, Bu-Ali Sina University, Hamedan 65174, I. R. Iran.

Accepted 21 December, 2012

Different common shapes of the RF-coil turns in induction heating systems were considered and corresponding results of electromagnetic fields, eddy currents and volumetric heat generation have been computed using a 2D finite element method. For calculations, rectangular (symmetric, asymmetric and shielded), circular, D-shape and -shape have been considered for the cross section of the coil turns. The obtained numerical results show the importance of cross section shape on the intensity and distribution of heat generation in the heating system. Key words: Computer simulation, induction heating, finite element analysis, modeling, metals.

INTRODUCTION Radio frequency induction heating is widely used in modern technologies of high temperature heat treatments of materials including surface and through hardening, annealing, quenching, tempering, melting and brazing. This method demands very accurate control of heating profile in depth and surface areas. On the other hand, behavior of most materials during the heat treatment is non-linear and very complex. It depends on several parameters such as magnetic permeability, thermo-mechanical properties, thermal history, applied frequency, and coil-work piece geometry and orientation. In general, the electromagnetic field distribution and heating pattern in a considered work piece depend implicitly on the coil geometry including the cross section shape of the rings. One of the most important aspects of an inductive heating system is the coil design (Leatherman and Stutz, 1969; Rudnev et al., 2003; Kang et al., 2003; Nemkov et al., 2005; Rudnev, 2008). Because the heating pattern reflects the inductor geometry, shape of the coil rings has a crucial effect among these factors. The coil should be coupled to the *Corresponding author. E-mail: mhtvkl@gmail.com, mht@basu.ac.ir. Tel: +98 811 8271541. Fax: +98 811 8280440.

PACS: 88.40.me, 02.70.Dh, 07.05.Tp, 81.05.Bx.

work piece part as closely as feasible for maximum energy transfer. It is desirable that the largest possible number of magnetic flux lines intersect the work piece at the area to be heated. A denser flux at this point will result in higher current generated in the part. A well-designed coil maintains a good coil efficiency and proper heating pattern.

The traditional methods based on experience and trial-and-error process for induction coil and process design are usually time consuming and costly. For this reason, computer modeling is one of the major factors of successful induction heating design. By using numerical simulation, induction heating process can be modeled, analyzed and optimized virtually without even requiring experimental verification for process where the material response and properties are known.

The goal of this article is to reveal the role of cross section shape of the RF-coil which turns on the induction heating process and compare the corresponding results of electromagnetic fields, Eddy currents and heat generation distribution using a 2D FEM numerical approach. In our previous studies (Tavakoli et al., 2010, 2011), attention was paid to the effects of the work piece, thickness and height on the heat generation in this system. But in the analysis presented here, the shape and location of the work piece is fixed and different coil cross section shapes are considered corresponding to the real heating situations.

Table 1. Operating parameters used for calculations.

Description (mm) Symbol Value

Workpiece radius rw 50

Workpiece height hw 186

Coil inner radius rco 85

Coil wall thickness lco 1

Height of coil turns hco 13

Distance between coil turns dco 3

MATHEMATICAL MODEL Governing equations The mathematical model of induction heating process applied for the calculations have been described in detail elsewhere (Tavakoli

Tavakoli et al. 49 et al., 2009; 2010). It can be summarized as follow. The assumptions are (1) The system is axi-symmetric. (2) All materials are isotropic and non-magnetic and have no net electric charge. (3) The displacement current is neglected. (4) The driving current in the RF-coil is time-harmonic and its distribution (also voltage) in the coil is uniform, i.e., we assume that the density of driving current in the coil cross sections is uniform. Under these assumptions, the governing equations are;

Jzrzrrr

BB

0

11

(1)

where

),,(),,( tzrrAtzrB

And

workpiecein thecurrent eddy

coil in thecurrent eddy and driving cos0

trJJ

trtJJJJ

JBw

eworkpiece

Bco

edcoil

(2)

with a solution of the form

tzrStzrCtzrB sin),(cos),(),,( (3)

and

workpiecein the )(2

coil-RF in the 2

),(

22

2

2

2

02

2

2

SCr

SrJ

Cr

zrq

w

co

co

(4)

where ),,( tzrB is the magnetic stream function, ( , , )A r z t

is the circumferential component of magnetic vector potential,

),( zrC and ),( zrS the in-phase and out-of-phase component,

respectively, ),( zrq the volumetric power generation,

frequency of the driving electrical current in the induction coil, J

the charge current density, the electrical conductivity of the

material and is only nonzero in conducting regions, 00 the

magnetic permeability of free space and t the time. The driving

current density in the induction coil is calculated by

)2/(0 NRVJ coilcoilco , where coilV is the total voltage of the

coil (Voltage regime), coilR is the mean value of the coil radius and

N is the number of coil turns. The boundary conditions

are 0B ; both in the far field zr, and at the axis of

symmetry 0r .

The set of fundamental equations with boundary conditions have been solved using finite element method.

The calculation conditions

Values of electrical conductivity employed for our calculations are

-17 m)-ohm(109.5 co (RF-coils) and

-16 m)-ohm(104w (work piece), magnetic

permeability is equal to H/m104 7

0 and operating

parameters are listed in Table 1. The considered system consists of a right cylindrical conductor load (that is, steel work piece) in the

direction of Oz which is surrounded by a multi turn cylindrical

induction coil (Figure 1). The induction coil has 6 hollow copper turns with the rectangular (symmetric, asymmetric and shielded), circular, D-shape and -shape with unique height and wall thickness (Figure 2).

The mathematical model presented in the previous section was solved by the finite element method (FEM) using the FlexPDE package (pdesolutions Inc). Particular attention was paid to the convergence of results in the dependence of the position of the artificial boundary and density of the discretization mesh. Very important was also appropriate unstructured triangular meshing of the investigated region to suppress numerical instabilities appearing during the computation process. The applied mesh consists of more than 6000 elements and 11500 nodes.

In order to compare the results of electromagnetic field and heat generation distribution, we have assumed a driving electrical

current with total voltage of vVcoil 200 and a frequency of

kHz1 in the RF-coil (typical values) for all cases. The results

based on this set of parameters will be presented now.

50 Sci. Res. Essays

Figure 1. Sketch of the induction heating setup including a right cylindrical workpiece and a 6-turn RF-coil.

Tavakoli et al. 51

Figure 2. The cross section shapes of the coil turns: (a) symmetric rectangular, (b) asymmetric rectangular, (c) shielded rectangular, (d) circular, (e) D-shape and (f) -shape. All cross sections have the same width (10 mm) except for the circular case that its diameter is equal to height of coil turns, but the thickness of all is 1 mm.

Figure 3. Components of the magnetic stream function ( B ) calculated for the symmetric

rectangular cross section shape of the coil turns. The left hand side shows the in-phase

component ( C ) with weberC 4

max 105.1 on the lowest and top edges of the RF-

coil and weberC 4

min 103.1 within the workpiece and close to the side wall. The

right hand side shows the out-of-phase component ( S ) with weberS 4

max 105.9

on the outer surfaces of the induction coil turns.

52 Sci. Res. Essays

Figure 4. Components of the magnetic stream function ( B ) calculated for the asymmetric rectangular

cross section shape of the coil turns. The left hand side shows the in-phase component ( C ) with

weberC 4

max 107.1 on the lowest and top edges of the RF-coil and

weberC 4

min 103.1 within the workpiece and close to the side wall. The right hand side shows

the out-of-phase component ( S ) with weberS 4

max 100.9 on the outer surfaces of the induction

coil turns.

RESULTS AND DISCUSSION Electromagnetic field Figures 3 to 8 show the distribution of in-phase component (right hand side) and out-of-phase component

(left hand side) of the magnetic stream function ( B ) for

the symmetric-, asymmetric-, and shielded rectangular, circular, D- and -shape of the coil cross section,

respectively. The maximum of in-phase component ( C )

is located at the lowest and top surfaces of the RF-coil while the minimum is located within the work piece and close to the middle part of the side wall. Variation of this component is too high in the area close to the maximum and minimum points. In other parts of the system, this component is nearly constant. For the out-of-phase

component ( S ), the maximum is located at the outer

surfaces of the induction coil turns and its intensity rapidly

decreases toward the work piece. This is a direct result of

the expulsion of the S -field component from within the

work piece body in all cases. The most important differences are as follow:

i) The intensity and distribution of C -field component is

similar except close to the RF-coil which is modified by the coil cross section shape. Only in the case of asymmetric rectangular shape (Figure 5), the corner effect is evident at the outer edges of the ends ring and for the applied frequency here.

ii) The S -field distribution has approximately linear

gradient in the space between the coil and the load side wall.

iii) The intensity and distribution of S -field component is

clearly similar in all cases except in the case of asymmetric rectangular shape (Figure 5) which is relatively weaker than other cases.

Tavakoli et al. 53

Figure 5. Components of the magnetic stream function (B ) calculated for the shielded

rectangular cross section shape of the coil turns. The left hand side shows the in-phase

component ( C ) with weberC 4

max 104.1 on the lowest and top edges of the RF-coil

and weberC 4

min 103.1 within the workpiece and close to the side wall. The right

hand side shows the out-of-phase component ( S ) with weberS 4

max 100.9 on the

outer surfaces of the induction coil turns.

Figure 6. Components of the magnetic stream function (B ) calculated for the circular

cross section shape of the coil turns. The left hand side shows the in-phase component

( C ) with weberC 4

max 105.1 on the lowest and top surfaces of the RF-coil and

weberC 4

min 102.1 within the workpiece and close to the side wall. The right

hand side shows the out-of-phase component ( S ) with weberS 4

max 105.9 on the

outer surfaces of the induction coil turns.

54 Sci. Res. Essays

Figure 7. Components of the magnetic stream function ( B ) calculated for the D-shape

cross section of the coil turns. The left hand side shows the in-phase component ( C ) with

weberC 4

max 106.1 on the lowest and top edges of the RF-coil and

weberC 4

min 102.1 within the workpiece and close to the side wall. The right

hand side shows the out-of-phase component ( S ) with weberS 4

max 105.9 on

the outer surfaces of the induction coil turns.

Figure 8. Components of the magnetic stream function ( B ) calculated for

the -shape cross section of the coil turns. The left hand side shows the in-

phase component ( C ) with weberC 4

max 104.2 on the lowest and

top edges of the RF-coil and weberC 4

min 100.2 within the

workpiece and close to the side wall. The right hand side shows the out-of-

phase component ( S ) with weberS 3

max 106.1 on the outer

surfaces of the induction coil turns.

Tavakoli et al. 55

Figure 9. Components of the eddy currents distribution in the workpiece calculated for the symmetric rectangular cross section shape of the coil turns. The left hand side shows the in-phase

component CJ and the right hand side shows the out-of-phase component SJ .

Eddy currents distribution Figures 9 to 14 represent the distribution of in-phase

component- CJ (right hand side) and out-of-phase

component- SJ (left hand side) of the eddy current for the

symmetric, asymmetric-, and shielded rectangular, circular, D- and -shape of the coil cross section, respectively. Both components of current density are non uniform along the work piece as well as its height due to non uniform distribution of the electromagnetic field components. The highest intensity of induced currents are produced within the area of the intense electromagnetic field components and so there is no induced current in the core of the load. At the side wall

surface Sc JJ 2 which means that within the

conductors the CJ component contributes to the heat

generation much more than the JS component. In addition, distribution and intensity of both components are similar within the work piece body and are always in the opposite direction to the driving current J0 on its side wall surface.

Volumetric heat generation The volumetric heat generation rate ( q ) in the work piece

has been shown in Figure 15 for all cases. The maximum value of energy deposition in the work piece load is located on the outer surface central part that faces the RF-coil and is decreased exponentially toward the core (skin effect). The structure of heating pattern of the work piece is caused by distribution and intensity of eddy currents components in that part of the system. The depth of heating is greater in the middle portion than at the ends of the load part. This heating profile depends strongly on the coil-work piece geometry and orientation. If the coil is lengthened until it overlaps the conductor part the heating will be more intense than at the ends than at the middle (Kang et al., 2003; Tavakoli et al., 2011). The heating profile at the side wall surface of the work piece is approximately uniform at the maximum value and then rapidly decreases toward the both ends (Figure 16) wherein the greater intensity of the asymmetric rectangular shape is clearly displayed. Also noteworthy is that the change in the coil cross section shape has not any effective influence on the spatial distribution of heat

56 Sci. Res. Essays

Figure 10. Components of the eddy currents distribution in the workpiece calculated for the asymmetric rectangular cross section shape of the coil turns. The left hand side

shows the in-phase component CJ and the right hand side shows the out-of-phase

component SJ .

Figure 11. Components of the eddy currents distribution in the workpiece calculated for the shielded rectangular cross section shape of the coil turns.

Tavakoli et al. 57

Figure 12. Components of the eddy currents distribution in the workpiece calculated for the circular cross section shape of the coil turns. The left hand side shows the in-phase

component CJ and the right hand side shows the out-of-phase component SJ .

Figure 13. Components of the eddy currents distribution in the workpiece calculated for the D-shape cross section of the coil turns. The left hand side shows the in-phase component

CJ and the right hand side shows the out-of-

phase component SJ .

58 Sci. Res. Essays

Figure 14. Components of the eddy currents distribution in the workpiece calculated for the -shape cross section of the coil turns. The left hand side shows the in-phase component

CJ and the

right hand side shows the out-of-phase component SJ .

Figure 15. Volumetric power distribution ( q ) in the work piece calculated

for the (a) symmetric rectangular, (b) asymmetric rectangular, (c) shielded rectangular, (d) circular, (e) D-shape and (f) -shape of the coil turns.

Tavakoli et al. 59

Figure 16. Profiles of the heat generated along the surface of the work piece side wall calculated for all configurations.

Table 2. Detail information about the heat generated in the heating setup, calculated for different cross section shapes of the RF-coil turns. (Efficiency =

100/ total

workpiece

total QQ ).

Cross section shape )(kWattQworkpiece

total )(kWattQcoil

total Efficiency

Circular 210.2 196.5 51.7

Symmetric rectangular 237.1 190.4 55.5

Asymmetric rectangular 243.7 186.9 56.6

Shielded rectangular 234.5 187.5 55.6

D-shape 218.1 211.9 50.1

-shape 199.9 184.1 52.1

generation in the work piece. It is necessary to mention here that the surface heating profile of the work piece is of great practical importance and depends on the applied frequency of input current, physical properties of the work piece, and the geometry and orientation of the coil-work piece (Leatherman and Stutz, 1969; Rudnev et al., 2003; Tavakoli et al., 2011). This heating profile affects the heat flow and temperature field of the work piece, directly (Kang et al., 2003; Tavakoli et al., 2011; Jung and Kang, 1999; Kang and Jung, 2000; Jung et al., 2000; Chen et al., 2004; Kawaguchi et al., 2005).

The spatial distribution of heat generation in the induction coil is mostly uniform with local "hot spots" (high concentrated power density) at the lowest and upper

edges, Figure 17. They are a result of the field curvature (especially C -field component) in tracing with the cross

section shape of coil turns. It shows the corner and edge effect which is a common occurrence in induction heating applications (Leatherman and Stutz, 1969; Rudnev et al., 2003; Tavakoli et al., 2010).

Another important parameter of an induction heating setup is heating efficiency. Heating efficiency is the percentage of the input power of the RF-coil that is transferred to the work piece by induction. In other words, it represents the ability of the RF-coil to induce power into the work piece. Detail information about the heat generated in the system and the heating efficiency of all cases considered here is shown in Table 2.

60 Sci. Res. Essays

Figure 17. Volumetric power distribution (q) in the RF-coil calculated the (a) symmetric rectangular, (b) asymmetric rectangular, (c) shielded rectangular, (d) circular, (e) D-shape and (f) -shape of the coil turns.

Conclusions In the present work, a two-dimensional induction heating problem, formulated in terms of stream function (vector potential) has been considered and the numerical calculation results of electromagnetic fields, eddy currents and generated power with different cross section shapes of the coil rings have been presented using a finite element method. The obtained results indicate that 1) The intensity and spatial distribution of all induction heating components are a complex function of the workpiece and RF-coil geometry including the cross section shape of the coil rings. 2) A correct choice of cross section shape of the coil turns leads to transfer of power more promptly to a load because of their ability to concentrate magnetic flux in the area to be heated. It allows the designers to avoid their undesirable heating and establish a suitable power structure. This is a possible way to control temperature distribution across the height and depth of the work piece, thermal deformation and electromagnetic forces on the load. 3) In base of this set of computations and considered frequency, the asymmetric rectangular cross section shape of the coil rings is the best for work piece power generation as well as the efficiency. REFERENCES Chen QS, Gao P, Hu WR (2004). Effects of induction heating on

temperature distribution and growth rate in large-size SiC growth system. J. Crystal Growth 266:320-326. http://www.pdesolutions.com/.

Jung HK, Kang CG (1999). An induction heating process with coil design and solutions avoiding coarsening phenomena of Al-6 Pct Si-3 Pct Cu-0.3 Pct Mg alloy for thixoforming. Metall. Mater. Trans. A. 30A:2967-2977.

Jung HK, Kang CG, Moon YH (2000). Induction heating of semisolid billet and control of globular microstructure to prevent coarsening phenomena. J. Mater. Eng. Perform. 9:12-234.

Kawaguchi H, Enokizono M, Todaka T (2005). Thermal and magnetic field analysis of induction heating problems. J. Mater. Process. Technol. 161:193-198.

Kang CG, Jung HK (2000). A study on a thixoforming process using the thixotropic behavior of an aluminum alloy with an equiaxed microstructure. J. Mater. Eng. Perform. 9:530-535.

Tavakoli et al. 61 Kang CG, Seo PK, Jung HK (2003). Numerical analysis by new

proposed coil design method in induction heating process for semi-solid forming and its experimental verification with globalization evaluation. Mater. Sci. Eng. A341:121-138.

Leatherman AF, Stutz DE (1969). Induction heating advances. National Aeronautics and Space Administration.

Nemkov V, Goldstein R, Bukanin V (2005). Optimal design of internal induction coils. Heat Treat. Prog. 5:85-89.

Rudnev V, Loveles D, Cook R, Black M (2003). Handbook of induction heating. New York NY.

Rudnev V (2008). Systematic analysis of induction coil failures, Part 11c: Frequency Selection. Heat Treat. Prog. 8:27-29.

Tavakoli MH, Karbaschi H, Samavat F (2009). Computational modeling of induction heating, Prog. Electromag. Res. Lett. 11:93-102.

Tavakoli MH, Karbaschi H, Samavat F (2010). Computational Study of Electromagnetic Fields, Eddy Currents and Induction Heating in Thin and Thick Workpieces. Com. Comput. Phys. 8:211-225.

Tavakoli MH, Karbaschi H, Samavat F (2011). Influence of work piece height on the induction heating process. Math. Computer Model. 54:50-58.