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Numerical and experimental thermal analysis for a metallic hollow cylinder subjected to step-wise electro-magnetic induction heating
Applied Thermal Engineering 27 (2007) 1883–1894 www.elsevier.com/locate/apthermeng
Numerical and experimental thermal analysis for a metallic hollow cylinder subjected to step-wise electro-magnetic induction heating Jiin-Yuh Jang *, Yu-Wei Chiu Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan Received 27 May 2006; accepted 30 December 2006 Available online 26 January 2007
Abstract In this study, basic electro-magnetic and heat transfer theories were applied to simulate the electro-magnetic and temperature fields in a steel hollow cylinder subjected to step-wise induction heating from outside. Three different sizes (Pipe A, Do · Di · L = 95 mm · 29 mm · 1000 mm, Pipe B, Do · Di · L = 110 mm · 39 mm · 1120 mm, Pipe C, Do · Di · L = 131 mm · 47 mm · 1450 mm) of the workpieces were numerically and experimentally investigated and compared. The temperatures on the inside and outside surface of the workpiece during the induction heating process were measured by thermocouples and an infrared thermal imaging system, respectively. The applied power input is a steep-wise function (constant high power, 0–8 min, and decrease to it 60%, 8–12 min, and then increase it original high power, 12–20 min). The process of induction heating heats the hollow cylinder from ambient temperature above the Curie point. It is shown that the inside temperature of the hollow cylinder is below the outside temperature initially (0–8 min), and then a constant temperature is held for approximately 4 min and finally the inside temperature is higher than the outside temperature. The numerical results agreed with the experimental data within 15%. The numerical simulation of three different air gaps (5 mm, 15 mm and 25 mm) between the coil and the workpiece were also performed. It is found that the temperature is increased as the air gap is decreased. The average temperatures of the hollow steel for air gap = 5 mm are 10 C and 15 C higher those for air gap = 15 mm, 25 mm, respectively. 2007 Elsevier Ltd. All rights reserved. Keywords: Numerical and experimental; Metallic hollow cylinder; Induction heating
1. Introduction The induction heating process has been widely applied in industrial operations. The basic principles of induction heating are Faraday’s and Ampere’s law. From these general laws of physics, it is demonstrated that an alternating voltage applied to the induction coil can produce an alternating magnetic flux, which produces an alternating voltage at the same frequency with the current of the coil. According to the Lentz’s law, the time-varying electro-magnetic field will induce the eddy current, which can generate a flux
*
Corresponding author. Tel.: +886 6 2088573; fax: +886 6 2342232. E-mail address: [email protected] (J.-Y. Jang).
1359-4311/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2006.12.025
opposite to the direction of the coil flux. The eddy current then produces heat by the Joule effect. Although induction heating has been successfully applied in many industry process such as induction metal melting, and it’s utilization in mold surface heating need to overcome several concerns including coil design, system operation and parameters control, etc. For many different purposes of the induction heating process, the design of the heating system could be complex and had to rely upon a trial and error process. Therefore, it is necessary to build a precise and suitable numerical simulation module for the investigation of the induction heating process. The first of the numerical techniques to be widely used for electro-heating problems was finite difference, and the method is still used today in certain applications [1].
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Nomenclature A magnetic vector potential (Wb m1) ^ A amplitude of magnetic vector potential (Wb m1) B magnetic flux density (Wb m2) Cp heat capacity of workpiece (J kg1 K1) D electric flux density (F V m2) E electric field intensity (V m1) f frequency (Hz) H magnetic field intensity (N Wb1) J total current density (A m2) Je eddy current density (A m2) k thermal conductivity (W m1 K1) q_ heat source (W m3) T temperature (K) T1 ambient temperature (K) X, Y, Z coordinates
er e0 / ^ / l lr l0 q r rb x n
relative electric permittivity (F m1) electric permittivity for free space (8.854187 · 1012 F m1) electric scalar potential (J C1) amplitude of electric scalar potential (J C1) magnetic permeability, l = lrl0 (Wb2 N1 m2) relative magnetic permeability (Wb2 N1 m2) magnetic permeability for free space (4p · 107 Wb2 N1 m2) resistivity (X m) electric conductivity (X1 m1) Stefan–Boltzmann constant (5.67 · 108 2 4 Wm K ) frequency emissivity
Greek symbols a electric charge density (C m3) e electric permittivity, e = ere0 (F m1)
Fig. 1. The physical model of workpiece and coil.
Aniserowicz et al. [2] presented a new algorithm using finite element method and which was a thermal postprocessing tool for the analysis of calcoil-system for a steel hollow cylinder. In this paper, a set of induction coils is distributed along the steel cylinder, and eddy currents induced on the surface of the cylinder cause intensive heating due to Joule’s law. The degree of skin effect depends on the frequency and material properties such as electrical resistivity and magnetic permeability of the billet. A pronounced skin effect would appear when a high frequency is applied during the induction heating process [3,4]. Nerg and Partanen [5] built a model for non-linear three-dimensional induction heating problems for a steel hollow cylinder. The model was based on the combination of linear and non-linear surface impedances evaluated using transient magnetic field calculation. The power dissipated in surface impedances was transferred to the thermal model as heat flux. However, because of the possible inaccuracies of the
Fig. 2. Computational domain.
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Inside diameter (mm) Outside diameter (mm) Length (mm) Weight (kg) Air gap between coil and workpiece (mm)
29 95 1000 50.75 15
39 110 1120 73.48 12.5
47 131 1450 34.45 17
material data, which accrue error to the calculation of surface impedance, the proposed approach is only suitable for practical inductor design work. Preston [6] described an economic three-dimensional solution using the concept of surface impedance to reduce the problem to a scalar potential formulation. The application of this method to the calculation of the rotor surface temperature of a solid pole synchronous generator under negative-sequence fault conditions showed satisfactory agreement with test measurement. In a study on inductive coil design, Kang et al. [7] presented four procedures to optimize the inductive coil design for the induction heating process. Effective coil length and coil inner diameter for the induction heating system were designed to minimize the electro-magnetic end effect. The induction heating coil was surrounding a billet and an adiabatic cover was set up on the top of the billet. Based on the results of inductive coil design, the minimum temperature difference between surface and center in eutectic melting could be reached when the optimal coil length was 180 mm. Urbanek et al. [8] assumed that magnetic permeability of the steel cylinder depending on both the magnetic field intensity (H) and the temperature (T) and proposed a method of solving magneto-thermal problems by means of a finite element algorithm using FLUX2D software. Simulations were made for the frequencies of 4, 16.5, and 25 kHz (I = 10 A) and a 3 mm inductor-cylinder gap. The differences between calculated and measured temperatures are less than 4%. Sadeghipour and Dopkin [9] employed a general-purpose finite element program to simulate and analyze high-frequency induction heating process for a steel hollow cylinder. The skin effect and the coupling procedure after the Curie temperature were investigated. The magnetic permeability would suddenly drop in any region where the temperature reaches the Curie point. Chaboudez et al. [10] dealt with numerical simulation of induction heating using axi-symmetric geometries. The sinusoidal voltage was imposed in the conductors. The choice of imposing the voltage was motivated by the better control of the voltage than the current. Three different coils have been used for the experiments, each of them having 25 windings and a length of 530 mm. Two of them have circular cross-sections, of inner diameters 74 mm and 64 mm, respectively. The cross-section of the third resembles a rectangle of 95 mm by 36 mm with round corners. Chen et al. [11] numerically and experimentally investigated the 3-D induction heating on a mold plate. A flat steel mold plate
600 I,V - T I V
500
500
400
400
300
300
200 0
b
600
200 120 240 360 480 600 720 840 960 1080 Time (Sec) 600
I,V - T I V
500
500
400
400
300
300
200 0
c
I (A)
Pipe C
600
I (A)
Pipe B
V (Voltage)
Pipe A
V (Voltage)
Size
200 120 240 360 480 600 720 840 960 1080 1200 Time (Sec)
800
800 I,V - T I V
700
V (Voltage)
1. 2. 3. 4. 5.
a
700
600
600
500
500
400
400
I (A)
Table 1 The geometrical data for the workpiece and coil
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300 0
300 120 240 360 480 600 720 840 960 1080 1200 Time (Sec)
Fig. 3. Applied voltage and current as a function of time for Pipes A, B and C.
of 220 mm by 160 mm by 15 mm is used, and circular (radius 63 mm) and rectangular coils (170 mm by 116 mm) were used to carry out the heating experiments. They found that multiple-turn coil provided better heating
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Fig. 4. Material conductivity and specific heat as a function of temperature: (a) conductivity and (b) specific heat.
efficiency and more uniformity in temperature distribution of the heated object. Preis [12] showed that the integro-differential method is more efficient than the superposition method when it is applied to multi-conductor systems with a high numbers of conductors and few finite elements within each conductor. Masse and Brevilies [13] introduced a prediction correction method, which gives a good initial set of values of the unknown at each time step and an automatic control of the time step according to the chosen accuracy. Muhlbauer et al. [14] introduced the boundary element method (BEM) to calculate the 3-D high-frequency electromagnetic fields. The set-up to be considered is divided into
Fig. 5. Material resistivity and relative magnetic permeability as a function of temperature: (a) resistivity and (b) relative magnetic permeability.
Table 2 Grid numbers Type Workpiece Coil Air Total grid numbers
Pipe A
Pipe B
Pipe C
506 72 930
600 72 940
966 72 940
1508
1612
1978
groups of bodies with specific material properties which allow the formulation of simplified conditional equations for electric vector potential at the surfaces of the set-up
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components. The physical fundamentals of the highfrequency electro-magnetic field connected with the specific advantages of the BEM, lead to a simulation program which runs on an efficient personal computer. This paper was inspired by the induction process for bi-metallic tubes. For the bonding of the metallic alloy powder (it is heated and melted around 800–900 C by induction heating) into the inner tube of a hollow steel cylinder, the resulting alloy layer may have defects of cavity and insufficient hardness. These can be attributed to the fact that applied step-wise power distributions are chosen improperly. In addition, the foregoing literature review shows that no related work on the numerical and experimental analysis for the induction heating subjected to a step-wise heating. This motivated the present investigation. The main purpose of this paper is to find out the optimal applied power distributions to have a constant temperature period between the inside and outside surfaces for the bonding process of a bi-metallic tube. In this paper, three different sizes of hollow steel cylinders (Pipe A, Do · Di · L = 95 mm · 29 mm · 1000 mm, Pipe B, Do · Di · L = 110 mm · 39 mm · 1120 mm, Pipe C, Do · Di · L = 131 mm · 47 mm · 1450 mm), as shown in Fig. 1, were investigated and compared during the induction heating process. The coupled thermal Fourier equation and electro-magnetic Maxwell equations were solved by the finite difference method. It will be shown that the applied power input is
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found to be a steep-wise function (constant high power, 0–8 min, and decrease to it 60%, 8–12 min, and then increase it original high power, 12–20 min). The numerical results are compared with experiment data to ensure the simulation model could precisely illustrate the actual situation during the induction heating process. 2. Mathematical analysis Fig. 2 designates the physical model and computational domain for the induction heating process for a hollow steel cylinder. Table 1 shows three different sizes of cylinders and coils for in this study (Pipe A, Do · Di · L = 95 mm · 29 mm · 1000 mm, Pipe B, Do · Di · L = 110 mm · 39 mm · 1120 mm, Pipe C, Do · Di · L = 131 mm · 47 mm · 1450 mm). The workpiece was heated by a power supply with step-wise voltage and current as a function of time as shown in Fig. 3. It was demonstrated by the experiments that there is no significant temperature variation along the axial direction of the cylinder. This allows one to reduce the 3-D field to a 2-D model. Therefore, two-dimensional physical models with radiation heat loss boundary conditions is suitable in this study. It is also assumed that the emissivity of the workpiece is constant. The properties of the workpiece and coil were assumed to be isotropic. For general time-varying electro-magnetic fields, Maxwell’s equations in differential from can be written as [15]
Fig. 6. Computational grid system.
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where r is the electric conductivity, e is the electric permittivity, and l is the magnetic permeability. Based on Eq. (2), the magnetic flux density B can be expressed in terms of a magnetic vector potential A as B = $ · A. It follows from Eq. (7) that oA ¼0 ð8Þ r Eþ ot By introducing the electric scalar potential /, which satisfies $ · $/ = 0, Eq. (8) can be integrated as follows: E¼
oA r/ ot
ð9Þ
Substituting B = $ · A into Eq. (4), one can obtain 1 2 oeE r A¼ rE l ot
ð10Þ
After substituting Eq. (9) into Eq. (10), the electro-magnetic field equation in terms of A and / is given below 1 2 o2 A or/ oA r A¼e 2 þe þr þ rr/ l ot ot ot
ð11Þ
2
where e oot2A þ e or/ is the displacement current density, r oA is ot ot the eddy current density, and r$/ (or J) is the conduction current density. For most practical application, the displacement current can be neglected, thus Eq. (11) can be simplified as 1 2 oA r AþJ r ¼0 l ot
ð12Þ
When electro-magnetic field quantities are harmonically oscillating functions with a single frequency f, and A can be expressed as
Fig. 7. Overall flowchart for the numerical simulation procedure.
rD¼a rB¼0 oB rE ¼ ot oD rH ¼J þ ot
ð1Þ ð2Þ ð3Þ ð4Þ
where D is the electric flux density, a is the electric charge density, B is the magnetic flux density, E is the electric field intensity, J is conduction current density and H is the magnetic field intensity. In addition, the following constitutive equations are holding true for a linear isotropic medium: J ¼ rE D ¼ eE
ð5Þ ð6Þ
B ¼ lH
ð7Þ
^ y; zÞejxt Aðx; y; z; tÞ ¼ Aðx; ð13Þ pffiffiffiffiffiffiffi ^ are amplitude of magnetic vector ^ and / where j ¼ 1, A potential and electric scalar potential respectively, x is the angle frequency (x = 2pf, f = 460 Hz in the present study). Substituting Eq. (13) into the above electro-magnetic field ^ equations, one can obtain the following equations for A: 1 2^ ^¼0 r A þ J jxrA l
ð14Þ
The time dependent heat transfer process in a steel hollow cylinder can be described by the Fourier equation. kðr2 T Þ þ q_ ¼ qcp
oT ot
ð15Þ
where T is temperature, k is thermal conductivity, q and Cp are the density and heat capacity, respectively, q_ is the heat source density induced by eddy current per unit time in a unit volume. The heat source density is related to conduction current density by Joule heating as shown below. !2 2 ^ jxt Þ J2 oA oðAe ^ jxt 2 ¼r ¼ r jxAe ð16Þ q_ ¼ ¼ r ot ot r
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Fig. 8. Experimental set-up.
Table 3 Three different air gaps for Pipe A
Air gap (mm)
Case 1
Case 2
Case 3
5
15
25
Eq. (15) is solved on the following boundary condition at the outside k
oT þ nrb ðT 4 T 41 Þ ¼ 0 on
ð17Þ
where T1 is the ambient temperature (K), which is set to be 298 K in this study, n is the normal vector, n is emissivity, and rb is the Stefan–Boltzmann constant (5.67 · 108 W m2 K4). It is noted that the convection boundary condition is neglected since its order is 3% compared to the radiation effect when there is a large difference in temperature between the workpiece and the ambient. An adiabatic boundary condition for the inside-surface was assumed, i.e., oT ¼0 on
ð18Þ
The physical properties (conductivity, specific heat, resistivity and relative magnetic permeability) of the workpiece are time-functions as shown in Figs. 4 and 5. 3. Numerical method The governing equations introduced above were solved numerically using a discrete control volume based on the finite difference formulation [16]. A proper grid system was necessary to obtained accurate solutions. According to the three models with different sizes, three different numbers of grids points (Pipe A = 1508, Pipe B = 1612 and
Pipe C = 1978), as shown in Table 2 and Fig. 6, were adopted in the computational domain. The electro-magnetic and thermal fields involve very different time scales. Since the electric current frequency typically used for induction heating is on the order 460 Hz, this results in a severely oscillating magnetic field. However, the timescale for the variation of temperature is on the order of seconds. For the initial temperature field, the electro-magnetic equation was solved using a very small time steps until a time-periodic solution is obtained. The computational results for the conduction current density were stored as the source terms to be used in the heat conduction equation, Eq. (15). The heat equation was solved by the finite volume method with a time interval about 1 s. After the new values of temperature and physical properties (conductivity, specific heat, resistivity and relative magnetic permeability) were obtained, the calculation proceeded to another computational time step of the electromagnetic field. This iteration was continued until the induction heating process is ended. Detailed flow chart is shown in Fig. 7. The computation was performed on an Intel Pentium4 3.0 GHz personal computer and the typical CPU times were 3–4 h for each case. 4. Experiment setup The experimental set-up is illustrated in Fig. 8. Three different sizes of hollow cylinders were tested in this study (Pipe A, Do · Di · L = 95 mm · 29 mm · 1000 mm, Pipe B, Do · Di · L = 110 mm · 39 mm · 1120 mm, Pipe C, Do · Di · L = 131 mm · 47 mm · 1450 mm). The power supply for induction heating experiment is the SCR (Silicon Controlled Rectifier) of Enercon Power Systems Incorporated, which could offer 180 Hz–10 kHz frequencies with power output 100–4000 K W. The workpiece
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was heated by a power supply with step-wise voltage and current as a function of time as shown in Fig. 3. To accurately measure and control the temperature of the workpieces, the K-type thermal couples were inserted into the hollow steel cylinder. Accuracy of the K-type thermal couples was approximately 0.2%. The temperatures on the outside-surface of the workpieces during
the induction heating process were measured by thermocouple and an infrared thermal imaging system, respectively. All the data signals were collected and converted by a data acquisition system (a hybrid recorder). The data acquisition system then transmitted the converted signals through a GPIB interface to the host computers.
Fig. 9. Magnetic flux density and eddy current distributions for Pipe A at different times: (a) at 100 s, (b) at 300 s and (c) at 800 s.
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Fig. 11. Temperature distributions (Pipe B) at 100, 500 and 1130 s. Fig. 10. Temperature distribution (Pipe A) at 100, 500 and 1060 s.
5. Results and discussion In this study, a simulation of three different sizes of hollow steel rod for the induction heating process was performed. Fig. 6 shows the magnetic flux density and the eddy current distribution on a cross-section of the workpiece during different times of Pipe A. From Fig. 9a the magnetic flux density on the area near the surface in the first 100 s is higher than the inside area, therefore, the eddy current is concentrated on the outside-surface. The depth of the thin contributed area is called the penetration depth [17]. During this time period, heat is generated in this area and trans-
ferred from the outside surface to the inside-surface. This phenomenon is called the skin effect. After 300 s, the differences of magnetic flux density and eddy current between the inside and the outside surfaces were less than that of the first 100 s. From Fig. 9b, it is clearly shown that the magnetic flux density and the eddy current on both sides of the cross-section of the workpiece are smaller than other area. This is because that the relative permeability (q) is decreasing as the temperature is increasing. At times = 800 s, the magnetic flux density and the eddy current were found spread uniformly across in the whole cross-section of the workpiece, as shown in Fig. 8c.
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1200 1100 1000
Temperature (°C)
900 800 700 600 500 Pipe A Experimental Outside T o Experimental Inside T i Numerical Outside T o Numerical Inside T i
400 300 200 100 0 0
120 240 360 480 600 720 840 960 1080 Time (s)
Fig. 13. Experimental and numerical average temperature distribution for Pipe A.
perature is 1056 C on the inside surface and the minimum is about 950 C on the outside-surface. Fig. 11 shows the temperature distribution of Pipe B at 100, 500 and 1130 s. From Table 1, the diameter of Pipe B is larger than pipe A’s, as result, more time was needed to reach the identical temperature comparing to Pipe A. Similarly, because Pipe C has the largest diameter of the three specimens, as shows in Table 1, the time required to reach the same temperature as Pipe A and Pipe B was proportionally longer, as evidently illustrated in Fig. 12a–c of the temperature distribution of Pipe C. Figs. 13–15 show a comparison between the temperature evolution curves obtained by measurement and numerical simulation. In order to discuss the temperature distribution, thermometers were positioned on both the inside-surface
1100 1000 Fig. 12. Temperature distributions (Pipe C) at 100, 500 and 1120 s.
900
Fig. 10 shows the temperature distribution of Pipe A at 100, 500 and 1060 s, respectively. In Fig. 10a, owing to the eddy current concentrated on both sides of workpiece near the coil, the temperature on both sides is higher than that of the central area. The maximum temperature is 363 C and the minimum is about 200 C. In Fig. 10b, the temperature distribution is the same as Fig. 10a. On account that the temperature reached the Curie point, the eddy current was uniformly distributed in the workpiece and the heat was transferred from the inside-surface to the outside-surface, resulting in higher temperature on the inside-surface. The maximum temperature in Fig. 10b is 857 C and the minimum is about 787 C. In Fig. 10c, the maximum tem-
Temperature (°C)
800 700 600 500 400
Pipe B Experimental Outside To Experimental Inside Ti Numerical Outside To Numerucal Inside Ti
300 200 100 0 0
120 240 360 480 600 720 840 960 1080 1200 Time (s)
Fig. 14. Experimental and numerical average temperature distribution for Pipe B.
J.-Y. Jang, Y.-W. Chiu / Applied Thermal Engineering 27 (2007) 1883–1894 1100 1000 900
Temperature (°C)
800 700 600 500 400 Pipe C Exper imental Exper imental Numerical Numerical
300 200 100
Outside T o Inside T i Outside To Inside T i
0 0
120 240 360 480 600 720 840 960 1080 1200 Time (s)
Fig. 15. Experimental and numerical average temperature distribution for Pipe C.
850
Temperature (°C)
800
750
Pipe A different gap Case 1 Case 1 Case 2 Case 2 Case 3 Case 3
700
650
600 300
360
420
480
540
Outside Surface To Inside Surface Ti Outside Surface To Inside Surface Ti Outside Surface To Inside Surface Ti
600
660
720
Times (s) Fig. 16. Average temperature distributions of Pipe A for three different air gaps (from 300 to 720 s).
and the outside-surface of the workpiece. It should be noted that before the temperature reached the Curie point, the temperature of the outside-surface is higher than the inside-surface. When the temperature reaches the Curie point, the supplied voltage was turned down to half of its initial value, so the temperature could be maintained at or above Curie point for a period of time (about 240 s). During this period, the inside-surface temperature caught and exceeded the outside-surface temperature. While turning down the input voltage during the induction heating process, as shown in Fig. 3, the eddy current spread uniformly on the workpiece, and the outside-surface cooled due to the heat exchanged with the environment. Therefore, during the later period of the induction heating process, the inside-
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surface temperature was higher than the outside-surface temperature. It is shown that the simulated temperature distributions are reasonable in agreement with the experimental data. For Pipe A, The maximum discrepancy between numerical and experimental results for Pipes A, B and C are 14.3%, 13.6% and 9.6%, respectively. The effect of the air gap between the coil and the workpiece on the temperature distribution is shown in Fig. 16. Table 3 shows 3 different air gaps in this study. It is found that the temperature is higher when the air gap is smaller. A close look of this figure indicated that the average temperatures of the hollow steel for air gap = 5 mm are 10 C and 15 C higher those for air gap = 15 mm, 25 mm, respectively. This is due to the fact that smaller air gap induces stronger magnetic field throughout the workpiece. 6. Conclusion The main purpose of this paper is to find what kind of applied power distributions can result in an isothermal distribution between the inside and outside-surface of a hollow cylinder for the bonding process of a bi-metallic tube. Three different sizes of the workpieces were numerically and experimentally investigated. The time variations of 2-D magnetic flux density, eddy current and temperature distribution during the induction heating process subjected to a step-wise function were presented. It was shown that the applied power input is found to be a steep-wise function (constant high power, 0–8 min, and decrease to it 60%, 8–12 min, and then increase it original high power, 12–20 min). The numerical results for the temperature distributions were reasonably in agreement with those of experimental data. The maximum discrepancy for Pipes A, B and C is 14.3%, 13.6% and 9.6%, respectively. It is also shown that the temperature is increased as the air gap is decreased. The average temperatures of the hollow steel for air gap = 5 mm are 10 C and 15 C higher those for air gap = 15 mm, 25 mm, respectively. Acknowledgement Financial support for this work was provided by the China Steel Corporations, Taiwan, under contract CSCRE93603. References [1] J.D. Lavers, Numerical solution methods for electroheating problems, IEEE Transactions on Magnetics 19 (6) (1983). [2] K. Aniserowicz, A. Skorek, C. Cossette, M.B. Zaremba, A new concept for finite element simulation of induction heating of steel cylinders, IEEE Transactions on Industry Applications 33 (4) (1997). [3] S.J. Salon, J.M. Schneider, A hybrid finite element-boundary integral formulation of the eddy current problem, IEEE Transactions on Magnetics 18 (2) (1982) 462. [4] T.F. Fawzi, K.F. Ali, P.E. Burke, Boundary integral equations analysis of induction devices with rotational symmetry, IEEE Transactions on Magnetics 19 (1) (1983).
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[5] J. Nerg, J. Partanen, A simplified FEM based calculation model for 3-D induction heating problems using surface impedance formulations, IEEE Transactions on Magnetics 37 (5) (2001). [6] T.W. Preston, An economic solution for 3-D coupled electromagnetic and thermal eddy current problems, IEEE Transactions on Magnetics 28 (2) (1992) 1992. [7] C.G. Kang, P.K. Seo, H.K. Jung, Numerical analysis by new proposed coil design method in induction heating process for semisolid forming and its experimental verification with globalization evaluation, Materials Science and Engineering A341 (2003) 121–138. [8] P. Urbanek, A. Skorek, M.B. Zaremba, Magnetic flux and temperature analysis in induction heated steel cylinder, IEEE Transactions on Magnetics 30 (5) (1994). [9] K. Sadeghipour, J.A. Dopkin, A computer aided finite element/ experimental analysis of induction heating process of steel, Computers in Industry 28 (1995) 195–205. [10] C. Chaboudez, D. Clain, R. Glardon, D. Mari, J. Rappaz, M. Swierkosz, Numerical modeling in induction heating for axisymmetric geometries, IEEE Transactions on Magnetics 33 (1) (1997).
[11] S.C. Chen, H.S. Peng, J.A. Chang, W.R. Jong, Simulations and verifications of induction heating on a mold plate, International Communications in Heat and Mass Transfer 31 (7) (2004) 971– 980. [12] K. Preis, A contribution to eddy current calculations in plan and axisymmetric multiconductor systems, IEEE Transactions on Magnetics Mag-19 (6) (1983). [13] P. Masse, T. Brevilies, A finite element prediction correction scheme for magneto-thermal coupled problem during curie transition, IEEE Transactions on Magnetics 21 (5) (1985). [14] A. Muhlbauer, A. Muiznieks, H.J. Lebmann, The calculation of 3D high-frequency electromagnetic fields during induction heating using the BEM, IEEE Transactions on Magnetics 29 (2) (2003). [15] J.C. Maxwell, A dynamic theory of the electromagnetic field, Royal Society Proceedings XIII (1864) 531–536. [16] S.V. Pantaker, A calculation procedure for two dimensional elliptic problem, Numeric Heat Transfer 4 (1981) 409–426. [17] E.J. Davies, Conduction and Induction Heating, Peter Peregrinus Ltd., London, United Kingdom, 1990, 1990.
Numerical and experimental thermal analysis for a metallic hollow cylinder subjected to step-wise electro-magnetic induction heating Jiin-Yuh Jang *, Yu-Wei Chiu Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan Received 27 May 2006; accepted 30 December 2006 Available online 26 January 2007
Abstract In this study, basic electro-magnetic and heat transfer theories were applied to simulate the electro-magnetic and temperature fields in a steel hollow cylinder subjected to step-wise induction heating from outside. Three different sizes (Pipe A, Do · Di · L = 95 mm · 29 mm · 1000 mm, Pipe B, Do · Di · L = 110 mm · 39 mm · 1120 mm, Pipe C, Do · Di · L = 131 mm · 47 mm · 1450 mm) of the workpieces were numerically and experimentally investigated and compared. The temperatures on the inside and outside surface of the workpiece during the induction heating process were measured by thermocouples and an infrared thermal imaging system, respectively. The applied power input is a steep-wise function (constant high power, 0–8 min, and decrease to it 60%, 8–12 min, and then increase it original high power, 12–20 min). The process of induction heating heats the hollow cylinder from ambient temperature above the Curie point. It is shown that the inside temperature of the hollow cylinder is below the outside temperature initially (0–8 min), and then a constant temperature is held for approximately 4 min and finally the inside temperature is higher than the outside temperature. The numerical results agreed with the experimental data within 15%. The numerical simulation of three different air gaps (5 mm, 15 mm and 25 mm) between the coil and the workpiece were also performed. It is found that the temperature is increased as the air gap is decreased. The average temperatures of the hollow steel for air gap = 5 mm are 10 C and 15 C higher those for air gap = 15 mm, 25 mm, respectively. 2007 Elsevier Ltd. All rights reserved. Keywords: Numerical and experimental; Metallic hollow cylinder; Induction heating
1. Introduction The induction heating process has been widely applied in industrial operations. The basic principles of induction heating are Faraday’s and Ampere’s law. From these general laws of physics, it is demonstrated that an alternating voltage applied to the induction coil can produce an alternating magnetic flux, which produces an alternating voltage at the same frequency with the current of the coil. According to the Lentz’s law, the time-varying electro-magnetic field will induce the eddy current, which can generate a flux
*
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1359-4311/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2006.12.025
opposite to the direction of the coil flux. The eddy current then produces heat by the Joule effect. Although induction heating has been successfully applied in many industry process such as induction metal melting, and it’s utilization in mold surface heating need to overcome several concerns including coil design, system operation and parameters control, etc. For many different purposes of the induction heating process, the design of the heating system could be complex and had to rely upon a trial and error process. Therefore, it is necessary to build a precise and suitable numerical simulation module for the investigation of the induction heating process. The first of the numerical techniques to be widely used for electro-heating problems was finite difference, and the method is still used today in certain applications [1].
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Nomenclature A magnetic vector potential (Wb m1) ^ A amplitude of magnetic vector potential (Wb m1) B magnetic flux density (Wb m2) Cp heat capacity of workpiece (J kg1 K1) D electric flux density (F V m2) E electric field intensity (V m1) f frequency (Hz) H magnetic field intensity (N Wb1) J total current density (A m2) Je eddy current density (A m2) k thermal conductivity (W m1 K1) q_ heat source (W m3) T temperature (K) T1 ambient temperature (K) X, Y, Z coordinates
er e0 / ^ / l lr l0 q r rb x n
relative electric permittivity (F m1) electric permittivity for free space (8.854187 · 1012 F m1) electric scalar potential (J C1) amplitude of electric scalar potential (J C1) magnetic permeability, l = lrl0 (Wb2 N1 m2) relative magnetic permeability (Wb2 N1 m2) magnetic permeability for free space (4p · 107 Wb2 N1 m2) resistivity (X m) electric conductivity (X1 m1) Stefan–Boltzmann constant (5.67 · 108 2 4 Wm K ) frequency emissivity
Greek symbols a electric charge density (C m3) e electric permittivity, e = ere0 (F m1)
Fig. 1. The physical model of workpiece and coil.
Aniserowicz et al. [2] presented a new algorithm using finite element method and which was a thermal postprocessing tool for the analysis of calcoil-system for a steel hollow cylinder. In this paper, a set of induction coils is distributed along the steel cylinder, and eddy currents induced on the surface of the cylinder cause intensive heating due to Joule’s law. The degree of skin effect depends on the frequency and material properties such as electrical resistivity and magnetic permeability of the billet. A pronounced skin effect would appear when a high frequency is applied during the induction heating process [3,4]. Nerg and Partanen [5] built a model for non-linear three-dimensional induction heating problems for a steel hollow cylinder. The model was based on the combination of linear and non-linear surface impedances evaluated using transient magnetic field calculation. The power dissipated in surface impedances was transferred to the thermal model as heat flux. However, because of the possible inaccuracies of the
Fig. 2. Computational domain.
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Inside diameter (mm) Outside diameter (mm) Length (mm) Weight (kg) Air gap between coil and workpiece (mm)
29 95 1000 50.75 15
39 110 1120 73.48 12.5
47 131 1450 34.45 17
material data, which accrue error to the calculation of surface impedance, the proposed approach is only suitable for practical inductor design work. Preston [6] described an economic three-dimensional solution using the concept of surface impedance to reduce the problem to a scalar potential formulation. The application of this method to the calculation of the rotor surface temperature of a solid pole synchronous generator under negative-sequence fault conditions showed satisfactory agreement with test measurement. In a study on inductive coil design, Kang et al. [7] presented four procedures to optimize the inductive coil design for the induction heating process. Effective coil length and coil inner diameter for the induction heating system were designed to minimize the electro-magnetic end effect. The induction heating coil was surrounding a billet and an adiabatic cover was set up on the top of the billet. Based on the results of inductive coil design, the minimum temperature difference between surface and center in eutectic melting could be reached when the optimal coil length was 180 mm. Urbanek et al. [8] assumed that magnetic permeability of the steel cylinder depending on both the magnetic field intensity (H) and the temperature (T) and proposed a method of solving magneto-thermal problems by means of a finite element algorithm using FLUX2D software. Simulations were made for the frequencies of 4, 16.5, and 25 kHz (I = 10 A) and a 3 mm inductor-cylinder gap. The differences between calculated and measured temperatures are less than 4%. Sadeghipour and Dopkin [9] employed a general-purpose finite element program to simulate and analyze high-frequency induction heating process for a steel hollow cylinder. The skin effect and the coupling procedure after the Curie temperature were investigated. The magnetic permeability would suddenly drop in any region where the temperature reaches the Curie point. Chaboudez et al. [10] dealt with numerical simulation of induction heating using axi-symmetric geometries. The sinusoidal voltage was imposed in the conductors. The choice of imposing the voltage was motivated by the better control of the voltage than the current. Three different coils have been used for the experiments, each of them having 25 windings and a length of 530 mm. Two of them have circular cross-sections, of inner diameters 74 mm and 64 mm, respectively. The cross-section of the third resembles a rectangle of 95 mm by 36 mm with round corners. Chen et al. [11] numerically and experimentally investigated the 3-D induction heating on a mold plate. A flat steel mold plate
600 I,V - T I V
500
500
400
400
300
300
200 0
b
600
200 120 240 360 480 600 720 840 960 1080 Time (Sec) 600
I,V - T I V
500
500
400
400
300
300
200 0
c
I (A)
Pipe C
600
I (A)
Pipe B
V (Voltage)
Pipe A
V (Voltage)
Size
200 120 240 360 480 600 720 840 960 1080 1200 Time (Sec)
800
800 I,V - T I V
700
V (Voltage)
1. 2. 3. 4. 5.
a
700
600
600
500
500
400
400
I (A)
Table 1 The geometrical data for the workpiece and coil
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300 0
300 120 240 360 480 600 720 840 960 1080 1200 Time (Sec)
Fig. 3. Applied voltage and current as a function of time for Pipes A, B and C.
of 220 mm by 160 mm by 15 mm is used, and circular (radius 63 mm) and rectangular coils (170 mm by 116 mm) were used to carry out the heating experiments. They found that multiple-turn coil provided better heating
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Fig. 4. Material conductivity and specific heat as a function of temperature: (a) conductivity and (b) specific heat.
efficiency and more uniformity in temperature distribution of the heated object. Preis [12] showed that the integro-differential method is more efficient than the superposition method when it is applied to multi-conductor systems with a high numbers of conductors and few finite elements within each conductor. Masse and Brevilies [13] introduced a prediction correction method, which gives a good initial set of values of the unknown at each time step and an automatic control of the time step according to the chosen accuracy. Muhlbauer et al. [14] introduced the boundary element method (BEM) to calculate the 3-D high-frequency electromagnetic fields. The set-up to be considered is divided into
Fig. 5. Material resistivity and relative magnetic permeability as a function of temperature: (a) resistivity and (b) relative magnetic permeability.
Table 2 Grid numbers Type Workpiece Coil Air Total grid numbers
Pipe A
Pipe B
Pipe C
506 72 930
600 72 940
966 72 940
1508
1612
1978
groups of bodies with specific material properties which allow the formulation of simplified conditional equations for electric vector potential at the surfaces of the set-up
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components. The physical fundamentals of the highfrequency electro-magnetic field connected with the specific advantages of the BEM, lead to a simulation program which runs on an efficient personal computer. This paper was inspired by the induction process for bi-metallic tubes. For the bonding of the metallic alloy powder (it is heated and melted around 800–900 C by induction heating) into the inner tube of a hollow steel cylinder, the resulting alloy layer may have defects of cavity and insufficient hardness. These can be attributed to the fact that applied step-wise power distributions are chosen improperly. In addition, the foregoing literature review shows that no related work on the numerical and experimental analysis for the induction heating subjected to a step-wise heating. This motivated the present investigation. The main purpose of this paper is to find out the optimal applied power distributions to have a constant temperature period between the inside and outside surfaces for the bonding process of a bi-metallic tube. In this paper, three different sizes of hollow steel cylinders (Pipe A, Do · Di · L = 95 mm · 29 mm · 1000 mm, Pipe B, Do · Di · L = 110 mm · 39 mm · 1120 mm, Pipe C, Do · Di · L = 131 mm · 47 mm · 1450 mm), as shown in Fig. 1, were investigated and compared during the induction heating process. The coupled thermal Fourier equation and electro-magnetic Maxwell equations were solved by the finite difference method. It will be shown that the applied power input is
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found to be a steep-wise function (constant high power, 0–8 min, and decrease to it 60%, 8–12 min, and then increase it original high power, 12–20 min). The numerical results are compared with experiment data to ensure the simulation model could precisely illustrate the actual situation during the induction heating process. 2. Mathematical analysis Fig. 2 designates the physical model and computational domain for the induction heating process for a hollow steel cylinder. Table 1 shows three different sizes of cylinders and coils for in this study (Pipe A, Do · Di · L = 95 mm · 29 mm · 1000 mm, Pipe B, Do · Di · L = 110 mm · 39 mm · 1120 mm, Pipe C, Do · Di · L = 131 mm · 47 mm · 1450 mm). The workpiece was heated by a power supply with step-wise voltage and current as a function of time as shown in Fig. 3. It was demonstrated by the experiments that there is no significant temperature variation along the axial direction of the cylinder. This allows one to reduce the 3-D field to a 2-D model. Therefore, two-dimensional physical models with radiation heat loss boundary conditions is suitable in this study. It is also assumed that the emissivity of the workpiece is constant. The properties of the workpiece and coil were assumed to be isotropic. For general time-varying electro-magnetic fields, Maxwell’s equations in differential from can be written as [15]
Fig. 6. Computational grid system.
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where r is the electric conductivity, e is the electric permittivity, and l is the magnetic permeability. Based on Eq. (2), the magnetic flux density B can be expressed in terms of a magnetic vector potential A as B = $ · A. It follows from Eq. (7) that oA ¼0 ð8Þ r Eþ ot By introducing the electric scalar potential /, which satisfies $ · $/ = 0, Eq. (8) can be integrated as follows: E¼
oA r/ ot
ð9Þ
Substituting B = $ · A into Eq. (4), one can obtain 1 2 oeE r A¼ rE l ot
ð10Þ
After substituting Eq. (9) into Eq. (10), the electro-magnetic field equation in terms of A and / is given below 1 2 o2 A or/ oA r A¼e 2 þe þr þ rr/ l ot ot ot
ð11Þ
2
where e oot2A þ e or/ is the displacement current density, r oA is ot ot the eddy current density, and r$/ (or J) is the conduction current density. For most practical application, the displacement current can be neglected, thus Eq. (11) can be simplified as 1 2 oA r AþJ r ¼0 l ot
ð12Þ
When electro-magnetic field quantities are harmonically oscillating functions with a single frequency f, and A can be expressed as
Fig. 7. Overall flowchart for the numerical simulation procedure.
rD¼a rB¼0 oB rE ¼ ot oD rH ¼J þ ot
ð1Þ ð2Þ ð3Þ ð4Þ
where D is the electric flux density, a is the electric charge density, B is the magnetic flux density, E is the electric field intensity, J is conduction current density and H is the magnetic field intensity. In addition, the following constitutive equations are holding true for a linear isotropic medium: J ¼ rE D ¼ eE
ð5Þ ð6Þ
B ¼ lH
ð7Þ
^ y; zÞejxt Aðx; y; z; tÞ ¼ Aðx; ð13Þ pffiffiffiffiffiffiffi ^ are amplitude of magnetic vector ^ and / where j ¼ 1, A potential and electric scalar potential respectively, x is the angle frequency (x = 2pf, f = 460 Hz in the present study). Substituting Eq. (13) into the above electro-magnetic field ^ equations, one can obtain the following equations for A: 1 2^ ^¼0 r A þ J jxrA l
ð14Þ
The time dependent heat transfer process in a steel hollow cylinder can be described by the Fourier equation. kðr2 T Þ þ q_ ¼ qcp
oT ot
ð15Þ
where T is temperature, k is thermal conductivity, q and Cp are the density and heat capacity, respectively, q_ is the heat source density induced by eddy current per unit time in a unit volume. The heat source density is related to conduction current density by Joule heating as shown below. !2 2 ^ jxt Þ J2 oA oðAe ^ jxt 2 ¼r ¼ r jxAe ð16Þ q_ ¼ ¼ r ot ot r
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Fig. 8. Experimental set-up.
Table 3 Three different air gaps for Pipe A
Air gap (mm)
Case 1
Case 2
Case 3
5
15
25
Eq. (15) is solved on the following boundary condition at the outside k
oT þ nrb ðT 4 T 41 Þ ¼ 0 on
ð17Þ
where T1 is the ambient temperature (K), which is set to be 298 K in this study, n is the normal vector, n is emissivity, and rb is the Stefan–Boltzmann constant (5.67 · 108 W m2 K4). It is noted that the convection boundary condition is neglected since its order is 3% compared to the radiation effect when there is a large difference in temperature between the workpiece and the ambient. An adiabatic boundary condition for the inside-surface was assumed, i.e., oT ¼0 on
ð18Þ
The physical properties (conductivity, specific heat, resistivity and relative magnetic permeability) of the workpiece are time-functions as shown in Figs. 4 and 5. 3. Numerical method The governing equations introduced above were solved numerically using a discrete control volume based on the finite difference formulation [16]. A proper grid system was necessary to obtained accurate solutions. According to the three models with different sizes, three different numbers of grids points (Pipe A = 1508, Pipe B = 1612 and
Pipe C = 1978), as shown in Table 2 and Fig. 6, were adopted in the computational domain. The electro-magnetic and thermal fields involve very different time scales. Since the electric current frequency typically used for induction heating is on the order 460 Hz, this results in a severely oscillating magnetic field. However, the timescale for the variation of temperature is on the order of seconds. For the initial temperature field, the electro-magnetic equation was solved using a very small time steps until a time-periodic solution is obtained. The computational results for the conduction current density were stored as the source terms to be used in the heat conduction equation, Eq. (15). The heat equation was solved by the finite volume method with a time interval about 1 s. After the new values of temperature and physical properties (conductivity, specific heat, resistivity and relative magnetic permeability) were obtained, the calculation proceeded to another computational time step of the electromagnetic field. This iteration was continued until the induction heating process is ended. Detailed flow chart is shown in Fig. 7. The computation was performed on an Intel Pentium4 3.0 GHz personal computer and the typical CPU times were 3–4 h for each case. 4. Experiment setup The experimental set-up is illustrated in Fig. 8. Three different sizes of hollow cylinders were tested in this study (Pipe A, Do · Di · L = 95 mm · 29 mm · 1000 mm, Pipe B, Do · Di · L = 110 mm · 39 mm · 1120 mm, Pipe C, Do · Di · L = 131 mm · 47 mm · 1450 mm). The power supply for induction heating experiment is the SCR (Silicon Controlled Rectifier) of Enercon Power Systems Incorporated, which could offer 180 Hz–10 kHz frequencies with power output 100–4000 K W. The workpiece
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was heated by a power supply with step-wise voltage and current as a function of time as shown in Fig. 3. To accurately measure and control the temperature of the workpieces, the K-type thermal couples were inserted into the hollow steel cylinder. Accuracy of the K-type thermal couples was approximately 0.2%. The temperatures on the outside-surface of the workpieces during
the induction heating process were measured by thermocouple and an infrared thermal imaging system, respectively. All the data signals were collected and converted by a data acquisition system (a hybrid recorder). The data acquisition system then transmitted the converted signals through a GPIB interface to the host computers.
Fig. 9. Magnetic flux density and eddy current distributions for Pipe A at different times: (a) at 100 s, (b) at 300 s and (c) at 800 s.
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Fig. 11. Temperature distributions (Pipe B) at 100, 500 and 1130 s. Fig. 10. Temperature distribution (Pipe A) at 100, 500 and 1060 s.
5. Results and discussion In this study, a simulation of three different sizes of hollow steel rod for the induction heating process was performed. Fig. 6 shows the magnetic flux density and the eddy current distribution on a cross-section of the workpiece during different times of Pipe A. From Fig. 9a the magnetic flux density on the area near the surface in the first 100 s is higher than the inside area, therefore, the eddy current is concentrated on the outside-surface. The depth of the thin contributed area is called the penetration depth [17]. During this time period, heat is generated in this area and trans-
ferred from the outside surface to the inside-surface. This phenomenon is called the skin effect. After 300 s, the differences of magnetic flux density and eddy current between the inside and the outside surfaces were less than that of the first 100 s. From Fig. 9b, it is clearly shown that the magnetic flux density and the eddy current on both sides of the cross-section of the workpiece are smaller than other area. This is because that the relative permeability (q) is decreasing as the temperature is increasing. At times = 800 s, the magnetic flux density and the eddy current were found spread uniformly across in the whole cross-section of the workpiece, as shown in Fig. 8c.
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1200 1100 1000
Temperature (°C)
900 800 700 600 500 Pipe A Experimental Outside T o Experimental Inside T i Numerical Outside T o Numerical Inside T i
400 300 200 100 0 0
120 240 360 480 600 720 840 960 1080 Time (s)
Fig. 13. Experimental and numerical average temperature distribution for Pipe A.
perature is 1056 C on the inside surface and the minimum is about 950 C on the outside-surface. Fig. 11 shows the temperature distribution of Pipe B at 100, 500 and 1130 s. From Table 1, the diameter of Pipe B is larger than pipe A’s, as result, more time was needed to reach the identical temperature comparing to Pipe A. Similarly, because Pipe C has the largest diameter of the three specimens, as shows in Table 1, the time required to reach the same temperature as Pipe A and Pipe B was proportionally longer, as evidently illustrated in Fig. 12a–c of the temperature distribution of Pipe C. Figs. 13–15 show a comparison between the temperature evolution curves obtained by measurement and numerical simulation. In order to discuss the temperature distribution, thermometers were positioned on both the inside-surface
1100 1000 Fig. 12. Temperature distributions (Pipe C) at 100, 500 and 1120 s.
900
Fig. 10 shows the temperature distribution of Pipe A at 100, 500 and 1060 s, respectively. In Fig. 10a, owing to the eddy current concentrated on both sides of workpiece near the coil, the temperature on both sides is higher than that of the central area. The maximum temperature is 363 C and the minimum is about 200 C. In Fig. 10b, the temperature distribution is the same as Fig. 10a. On account that the temperature reached the Curie point, the eddy current was uniformly distributed in the workpiece and the heat was transferred from the inside-surface to the outside-surface, resulting in higher temperature on the inside-surface. The maximum temperature in Fig. 10b is 857 C and the minimum is about 787 C. In Fig. 10c, the maximum tem-
Temperature (°C)
800 700 600 500 400
Pipe B Experimental Outside To Experimental Inside Ti Numerical Outside To Numerucal Inside Ti
300 200 100 0 0
120 240 360 480 600 720 840 960 1080 1200 Time (s)
Fig. 14. Experimental and numerical average temperature distribution for Pipe B.
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Temperature (°C)
800 700 600 500 400 Pipe C Exper imental Exper imental Numerical Numerical
300 200 100
Outside T o Inside T i Outside To Inside T i
0 0
120 240 360 480 600 720 840 960 1080 1200 Time (s)
Fig. 15. Experimental and numerical average temperature distribution for Pipe C.
850
Temperature (°C)
800
750
Pipe A different gap Case 1 Case 1 Case 2 Case 2 Case 3 Case 3
700
650
600 300
360
420
480
540
Outside Surface To Inside Surface Ti Outside Surface To Inside Surface Ti Outside Surface To Inside Surface Ti
600
660
720
Times (s) Fig. 16. Average temperature distributions of Pipe A for three different air gaps (from 300 to 720 s).
and the outside-surface of the workpiece. It should be noted that before the temperature reached the Curie point, the temperature of the outside-surface is higher than the inside-surface. When the temperature reaches the Curie point, the supplied voltage was turned down to half of its initial value, so the temperature could be maintained at or above Curie point for a period of time (about 240 s). During this period, the inside-surface temperature caught and exceeded the outside-surface temperature. While turning down the input voltage during the induction heating process, as shown in Fig. 3, the eddy current spread uniformly on the workpiece, and the outside-surface cooled due to the heat exchanged with the environment. Therefore, during the later period of the induction heating process, the inside-
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surface temperature was higher than the outside-surface temperature. It is shown that the simulated temperature distributions are reasonable in agreement with the experimental data. For Pipe A, The maximum discrepancy between numerical and experimental results for Pipes A, B and C are 14.3%, 13.6% and 9.6%, respectively. The effect of the air gap between the coil and the workpiece on the temperature distribution is shown in Fig. 16. Table 3 shows 3 different air gaps in this study. It is found that the temperature is higher when the air gap is smaller. A close look of this figure indicated that the average temperatures of the hollow steel for air gap = 5 mm are 10 C and 15 C higher those for air gap = 15 mm, 25 mm, respectively. This is due to the fact that smaller air gap induces stronger magnetic field throughout the workpiece. 6. Conclusion The main purpose of this paper is to find what kind of applied power distributions can result in an isothermal distribution between the inside and outside-surface of a hollow cylinder for the bonding process of a bi-metallic tube. Three different sizes of the workpieces were numerically and experimentally investigated. The time variations of 2-D magnetic flux density, eddy current and temperature distribution during the induction heating process subjected to a step-wise function were presented. It was shown that the applied power input is found to be a steep-wise function (constant high power, 0–8 min, and decrease to it 60%, 8–12 min, and then increase it original high power, 12–20 min). The numerical results for the temperature distributions were reasonably in agreement with those of experimental data. The maximum discrepancy for Pipes A, B and C is 14.3%, 13.6% and 9.6%, respectively. It is also shown that the temperature is increased as the air gap is decreased. The average temperatures of the hollow steel for air gap = 5 mm are 10 C and 15 C higher those for air gap = 15 mm, 25 mm, respectively. Acknowledgement Financial support for this work was provided by the China Steel Corporations, Taiwan, under contract CSCRE93603. References [1] J.D. Lavers, Numerical solution methods for electroheating problems, IEEE Transactions on Magnetics 19 (6) (1983). [2] K. Aniserowicz, A. Skorek, C. Cossette, M.B. Zaremba, A new concept for finite element simulation of induction heating of steel cylinders, IEEE Transactions on Industry Applications 33 (4) (1997). [3] S.J. Salon, J.M. Schneider, A hybrid finite element-boundary integral formulation of the eddy current problem, IEEE Transactions on Magnetics 18 (2) (1982) 462. [4] T.F. Fawzi, K.F. Ali, P.E. Burke, Boundary integral equations analysis of induction devices with rotational symmetry, IEEE Transactions on Magnetics 19 (1) (1983).
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[5] J. Nerg, J. Partanen, A simplified FEM based calculation model for 3-D induction heating problems using surface impedance formulations, IEEE Transactions on Magnetics 37 (5) (2001). [6] T.W. Preston, An economic solution for 3-D coupled electromagnetic and thermal eddy current problems, IEEE Transactions on Magnetics 28 (2) (1992) 1992. [7] C.G. Kang, P.K. Seo, H.K. Jung, Numerical analysis by new proposed coil design method in induction heating process for semisolid forming and its experimental verification with globalization evaluation, Materials Science and Engineering A341 (2003) 121–138. [8] P. Urbanek, A. Skorek, M.B. Zaremba, Magnetic flux and temperature analysis in induction heated steel cylinder, IEEE Transactions on Magnetics 30 (5) (1994). [9] K. Sadeghipour, J.A. Dopkin, A computer aided finite element/ experimental analysis of induction heating process of steel, Computers in Industry 28 (1995) 195–205. [10] C. Chaboudez, D. Clain, R. Glardon, D. Mari, J. Rappaz, M. Swierkosz, Numerical modeling in induction heating for axisymmetric geometries, IEEE Transactions on Magnetics 33 (1) (1997).
[11] S.C. Chen, H.S. Peng, J.A. Chang, W.R. Jong, Simulations and verifications of induction heating on a mold plate, International Communications in Heat and Mass Transfer 31 (7) (2004) 971– 980. [12] K. Preis, A contribution to eddy current calculations in plan and axisymmetric multiconductor systems, IEEE Transactions on Magnetics Mag-19 (6) (1983). [13] P. Masse, T. Brevilies, A finite element prediction correction scheme for magneto-thermal coupled problem during curie transition, IEEE Transactions on Magnetics 21 (5) (1985). [14] A. Muhlbauer, A. Muiznieks, H.J. Lebmann, The calculation of 3D high-frequency electromagnetic fields during induction heating using the BEM, IEEE Transactions on Magnetics 29 (2) (2003). [15] J.C. Maxwell, A dynamic theory of the electromagnetic field, Royal Society Proceedings XIII (1864) 531–536. [16] S.V. Pantaker, A calculation procedure for two dimensional elliptic problem, Numeric Heat Transfer 4 (1981) 409–426. [17] E.J. Davies, Conduction and Induction Heating, Peter Peregrinus Ltd., London, United Kingdom, 1990, 1990.