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, respec-
tively. 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 opposite to the direction of the coil flux. The eddy current
then produces heat by the Joule effect. Although induction
The induction heating process has been widely applied in heating has been successfully applied in many industry pro-
industrial operations. The basic principles of induction cess such as induction metal melting, and it’s utilization in
heating are Faraday’s and Ampere’s law. From these gen- mold surface heating need to overcome several concerns
eral laws of physics, it is demonstrated that an alternating including coil design, system operation and parameters con-
voltage applied to the induction coil can produce an alter- trol, etc. For many different purposes of the induction heat-
nating magnetic flux, which produces an alternating voltage ing process, the design of the heating system could be
at the same frequency with the current of the coil. Accord- complex and had to rely upon a trial and error process.
ing to the Lentz’s law, the time-varying electro-magnetic Therefore, it is necessary to build a precise and suitable
field will induce the eddy current, which can generate a flux numerical simulation module for the investigation of the
induction heating process. The first of the numerical tech-
niques to be widely used for electro-heating problems was
*
Corresponding author. Tel.: +886 6 2088573; fax: +886 6 2342232. finite difference, and the method is still used today in certain
E-mail address: jangjim@mail.ncku.edu.tw (J.-Y. Jang). applications [1].

1359-4311/$ - see front matter  2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.applthermaleng.2006.12.025
1884 J.-Y. Jang, Y.-W. Chiu / Applied Thermal Engineering 27 (2007) 1883–1894

Nomenclature

A magnetic vector potential (Wb m1) er relative electric permittivity (F m1)

A^ amplitude of magnetic vector potential (Wb m1) e0 electric permittivity for free space
B magnetic flux density (Wb m2) (8.854187 · 1012 F m1)
Cp heat capacity of workpiece (J kg1 K1) / electric scalar potential (J C1)
D electric flux density (F V m2) ^
/ amplitude of electric scalar potential (J C1)
E electric field intensity (V m1) l magnetic permeability, l = lrl0 (Wb2 N1 m2)
f frequency (Hz) lr relative magnetic permeability (Wb2 N1 m2)
H magnetic field intensity (N Wb1) l0 magnetic permeability for free space (4p ·
J total current density (A m2) 107 Wb2 N1 m2)
Je eddy current density (A m2) q resistivity (X m)
k thermal conductivity (W m1 K1) r electric conductivity (X1 m1)
q_ heat source (W m3) rb Stefan–Boltzmann constant (5.67 · 108
2 4
T temperature (K) Wm K )
T1 ambient temperature (K) x frequency
X, Y, Z coordinates n emissivity

Greek symbols
a electric charge density (C m3)
e electric permittivity, e = ere0 (F m1)

Aniserowicz et al. [2] presented a new algorithm using

finite element method and which was a thermal post-
processing tool for the analysis of calcoil-system for a steel
hollow cylinder. In this paper, a set of induction coils is dis-
tributed 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 fre-
quency 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 dur-
ing the induction heating process [3,4]. Nerg and Partanen
[5] built a model for non-linear three-dimensional induc-
tion heating problems for a steel hollow cylinder. The
model was based on the combination of linear and non-lin-
ear surface impedances evaluated using transient magnetic
field calculation. The power dissipated in surface imped-
ances was transferred to the thermal model as heat
Fig. 1. The physical model of workpiece and coil. flux. However, because of the possible inaccuracies of the

Fig. 2. Computational domain.

 J.-Y. Jang, Y.-W. Chiu / Applied Thermal Engineering 27 (2007) 1883–1894 1885

Table 1
The geometrical data for the workpiece and coil a 600 600
I,V - T
Size Pipe A Pipe B Pipe C I
1. Inside diameter (mm) 29 39 47 V
500 500
2. Outside diameter (mm) 95 110 131
3. Length (mm) 1000 1120 1450

V (Voltage)
4. Weight (kg) 50.75 73.48 34.45
5. Air gap between coil and workpiece 15 12.5 17

I (A)
400 400
(mm)

material data, which accrue error to the calculation of sur- 300 300
face impedance, the proposed approach is only suitable for
practical inductor design work. Preston [6] described an
economic three-dimensional solution using the concept of 200 200
surface impedance to reduce the problem to a scalar poten- 0 120 240 360 480 600 720 840 960 1080
Time (Sec)
tial formulation. The application of this method to the cal-
culation of the rotor surface temperature of a solid pole 600 600
synchronous generator under negative-sequence fault con- b I,V - T
I
ditions showed satisfactory agreement with test measure- V
ment. In a study on inductive coil design, Kang et al. [7]
500 500
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 V (Voltage)

I (A)
system were designed to minimize the electro-magnetic 400 400
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 min- 300 300
imum temperature difference between surface and center in
eutectic melting could be reached when the optimal coil
length was 180 mm. 200 200
Urbanek et al. [8] assumed that magnetic permeability of 0 120 240 360 480 600 720 840 960 1080 1200
the steel cylinder depending on both the magnetic field Time (Sec)

intensity (H) and the temperature (T) and proposed a

c 800 800
method of solving magneto-thermal problems by means I,V - T
I
of a finite element algorithm using FLUX2D software.
V
Simulations were made for the frequencies of 4, 16.5, and 700 700
25 kHz (I = 10 A) and a 3 mm inductor-cylinder gap. The
differences between calculated and measured temperatures
600 600
V (Voltage)

are less than 4%. Sadeghipour and Dopkin [9] employed

I (A)
a general-purpose finite element program to simulate and
analyze high-frequency induction heating process for a 500 500
steel hollow cylinder. The skin effect and the coupling pro-
cedure after the Curie temperature were investigated. The
magnetic permeability would suddenly drop in any region 400 400
where the temperature reaches the Curie point.
Chaboudez et al. [10] dealt with numerical simulation of
300 300
induction heating using axi-symmetric geometries. The 0 120 240 360 480 600 720 840 960 1080 1200
sinusoidal voltage was imposed in the conductors. The Time (Sec)
choice of imposing the voltage was motivated by the better Fig. 3. Applied voltage and current as a function of time for Pipes A, B
control of the voltage than the current. Three different coils and C.
have been used for the experiments, each of them having 25
windings and a length of 530 mm. Two of them have circu-
lar cross-sections, of inner diameters 74 mm and 64 mm,
respectively. The cross-section of the third resembles a rect- of 220 mm by 160 mm by 15 mm is used, and circular
angle of 95 mm by 36 mm with round corners. Chen et al. (radius 63 mm) and rectangular coils (170 mm by
[11] numerically and experimentally investigated the 3-D 116 mm) were used to carry out the heating experiments.
induction heating on a mold plate. A flat steel mold plate They found that multiple-turn coil provided better heating
1886 J.-Y. Jang, Y.-W. Chiu / Applied Thermal Engineering 27 (2007) 1883–1894

Fig. 5. Material resistivity and relative magnetic permeability as a

Fig. 4. Material conductivity and specific heat as a function of temper- function of temperature: (a) resistivity and (b) relative magnetic
ature: (a) conductivity and (b) specific heat. permeability.

efficiency and more uniformity in temperature distribution

of the heated object. Table 2
Preis [12] showed that the integro-differential method is Grid numbers
more efficient than the superposition method when it is Type Pipe A Pipe B Pipe C
applied to multi-conductor systems with a high numbers Workpiece 506 600 966
of conductors and few finite elements within each conduc- Coil 72 72 72
tor. Masse and Brevilies [13] introduced a prediction cor- Air 930 940 940
rection method, which gives a good initial set of values of Total grid numbers 1508 1612 1978
the unknown at each time step and an automatic control
of the time step according to the chosen accuracy. Muhl-
bauer et al. [14] introduced the boundary element method groups of bodies with specific material properties which
(BEM) to calculate the 3-D high-frequency electro- allow the formulation of simplified conditional equations
magnetic fields. The set-up to be considered is divided into for electric vector potential at the surfaces of the set-up
 J.-Y. Jang, Y.-W. Chiu / Applied Thermal Engineering 27 (2007) 1883–1894 1887

components. The physical fundamentals of the high- found to be a steep-wise function (constant high power,
frequency electro-magnetic field connected with the specific 0–8 min, and decrease to it 60%, 8–12 min, and then
advantages of the BEM, lead to a simulation program increase it original high power, 12–20 min). The numerical
which runs on an efficient personal computer. results are compared with experiment data to ensure the
This paper was inspired by the induction process for simulation model could precisely illustrate the actual situa-
bi-metallic tubes. For the bonding of the metallic alloy tion during the induction heating process.
powder (it is heated and melted around 800–900 C by
induction heating) into the inner tube of a hollow steel cyl- 2. Mathematical analysis
inder, the resulting alloy layer may have defects of cavity
and insufficient hardness. These can be attributed to the Fig. 2 designates the physical model and computational
fact that applied step-wise power distributions are chosen domain for the induction heating process for a hollow steel
improperly. In addition, the foregoing literature review cylinder. Table 1 shows three different sizes of cylinders
shows that no related work on the numerical and experi- and coils for in this study (Pipe A, Do · Di · L = 95 mm ·
mental analysis for the induction heating subjected to a 29 mm · 1000 mm, Pipe B, Do · Di · L = 110 mm · 39 mm ·
step-wise heating. This motivated the present investigation. 1120 mm, Pipe C, Do · Di · L = 131 mm · 47 mm ·
The main purpose of this paper is to find out the optimal 1450 mm). The workpiece was heated by a power supply
applied power distributions to have a constant temperature with step-wise voltage and current as a function of time
period between the inside and outside surfaces for the as shown in Fig. 3. It was demonstrated by the experiments
bonding process of a bi-metallic tube. In this paper, three that there is no significant temperature variation along the
different sizes of hollow steel cylinders (Pipe A, Do · Di · axial direction of the cylinder. This allows one to reduce
L = 95 mm · 29 mm · 1000 mm, Pipe B, Do · Di · L = the 3-D field to a 2-D model. Therefore, two-dimensional
110 mm · 39 mm · 1120 mm, Pipe C, Do · Di · L = 131 mm · physical models with radiation heat loss boundary condi-
47 mm · 1450 mm), as shown in Fig. 1, were investigated tions is suitable in this study. It is also assumed that the
and compared during the induction heating process. The emissivity of the workpiece is constant. The properties of
coupled thermal Fourier equation and electro-magnetic the workpiece and coil were assumed to be isotropic.
Maxwell equations were solved by the finite difference For general time-varying electro-magnetic fields, Max-
method. It will be shown that the applied power input is well’s equations in differential from can be written as [15]

Fig. 6. Computational grid system.

1888 J.-Y. Jang, Y.-W. Chiu / Applied Thermal Engineering 27 (2007) 1883–1894

where r is the electric conductivity, e is the electric permit-

tivity, 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
r Eþ ¼0 ð8Þ
ot
By introducing the electric scalar potential /, which satis-
fies $ · $/ = 0, Eq. (8) can be integrated as follows:
oA
E¼  r/ ð9Þ
ot
Substituting B = $ · A into Eq. (4), one can obtain
1 2 oeE
r A¼  rE ð10Þ
l ot
After substituting Eq. (9) into Eq. (10), the electro-mag-
netic field equation in terms of A and / is given below
1 2 o2 A or/ oA
r A¼e 2 þe þr þ rr/ ð11Þ
l ot ot ot
2
where e oot2A þ e or/
ot
is the displacement current density, r oA
ot
is
the eddy current density, and r$/ (or J) is the conduction
current density. For most practical application, the dis-
placement current can be neglected, thus Eq. (11) can be
simplified as
1 2 oA
r AþJ r ¼0 ð12Þ
l ot
When electro-magnetic field quantities are harmonically
oscillating functions with a single frequency f, and A can be
expressed as
^ y; zÞejxt
Aðx; y; z; tÞ ¼ Aðx; ð13Þ
pffiffiffiffiffiffiffi
where j ¼ 1, A ^ are amplitude of magnetic vector
^ and /
Fig. 7. Overall flowchart for the numerical simulation procedure.
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
rD¼a ð1Þ equations, one can obtain the following equations for A: ^
rB¼0 ð2Þ 1 2^ ^¼0
r A þ J  jxrA ð14Þ
oB l
rE ¼ ð3Þ
ot
oD The time dependent heat transfer process in a steel hol-
rH ¼J þ ð4Þ low cylinder can be described by the Fourier equation.
ot
oT
where D is the electric flux density, a is the electric charge kðr2 T Þ þ q_ ¼ qcp ð15Þ
density, B is the magnetic flux density, E is the electric field ot
intensity, J is conduction current density and H is the where T is temperature, k is thermal conductivity, q and Cp
magnetic field intensity. In addition, the following constitu- are the density and heat capacity, respectively, q_ is the heat
tive equations are holding true for a linear isotropic source density induced by eddy current per unit time in a
medium: unit volume. The heat source density is related to conduc-
tion current density by Joule heating as shown below.
J ¼ rE ð5Þ !2
 2 ^ jxt Þ
D ¼ eE ð6Þ J2 oA oðAe  
q_ ¼ ¼ r ¼r ^ jxt 2
¼ r jxAe ð16Þ
B ¼ lH ð7Þ r ot ot
 J.-Y. Jang, Y.-W. Chiu / Applied Thermal Engineering 27 (2007) 1883–1894 1889

Fig. 8. Experimental set-up.

Table 3 Pipe C = 1978), as shown in Table 2 and Fig. 6, were

Three different air gaps for Pipe A adopted in the computational domain.
Case 1 Case 2 Case 3 The electro-magnetic and thermal fields involve very dif-
Air gap (mm) 5 15 25 ferent time scales. Since the electric current frequency typ-
ically used for induction heating is on the order 460 Hz,
this results in a severely oscillating magnetic field. How-
Eq. (15) is solved on the following boundary condition ever, the timescale for the variation of temperature is on
at the outside the order of seconds. For the initial temperature field, the
oT electro-magnetic equation was solved using a very small
k þ nrb ðT 4  T 41 Þ ¼ 0 ð17Þ time steps until a time-periodic solution is obtained. The
on
computational results for the conduction current density
where T1 is the ambient temperature (K), which is set to be were stored as the source terms to be used in the heat con-
298 K in this study, n is the normal vector, n is emissivity, duction equation, Eq. (15). The heat equation was solved
and rb is the Stefan–Boltzmann constant (5.67 · 108 by the finite volume method with a time interval about
W m2 K4). It is noted that the convection boundary con- 1 s. After the new values of temperature and physical prop-
dition is neglected since its order is 3% compared to the erties (conductivity, specific heat, resistivity and relative
radiation effect when there is a large difference in tempera- magnetic permeability) were obtained, the calculation pro-
ture between the workpiece and the ambient. An adiabatic ceeded to another computational time step of the electro-
boundary condition for the inside-surface was assumed, magnetic field. This iteration was continued until the
i.e., induction heating process is ended. Detailed flow chart is
oT shown in Fig. 7. The computation was performed on an
¼0 ð18Þ Intel Pentium4 3.0 GHz personal computer and the typical
on
CPU times were 3–4 h for each case.
The physical properties (conductivity, specific heat,
resistivity and relative magnetic permeability) of the work- 4. Experiment setup
piece are time-functions as shown in Figs. 4 and 5.
The experimental set-up is illustrated in Fig. 8. Three
3. Numerical method different sizes of hollow cylinders were tested in this study
(Pipe A, Do · Di · L = 95 mm · 29 mm · 1000 mm, Pipe
The governing equations introduced above were solved B, Do · Di · L = 110 mm · 39 mm · 1120 mm, Pipe C,
numerically using a discrete control volume based on the Do · Di · L = 131 mm · 47 mm · 1450 mm). The power
finite difference formulation [16]. A proper grid system supply for induction heating experiment is the SCR
was necessary to obtained accurate solutions. According (Silicon Controlled Rectifier) of Enercon Power Systems
to the three models with different sizes, three different num- Incorporated, which could offer 180 Hz–10 kHz frequen-
bers of grids points (Pipe A = 1508, Pipe B = 1612 and cies with power output 100–4000 K W. The workpiece
1890 J.-Y. Jang, Y.-W. Chiu / Applied Thermal Engineering 27 (2007) 1883–1894

was heated by a power supply with step-wise voltage and the induction heating process were measured by
current as a function of time as shown in Fig. 3. thermocouple and an infrared thermal imaging system,
To accurately measure and control the temperature of respectively. All the data signals were collected and con-
the workpieces, the K-type thermal couples were inserted verted by a data acquisition system (a hybrid recorder).
into the hollow steel cylinder. Accuracy of the K-type The data acquisition system then transmitted the con-
thermal couples was approximately 0.2%. The tempera- verted signals through a GPIB interface to the host
tures on the outside-surface of the workpieces during 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.
 J.-Y. Jang, Y.-W. Chiu / Applied Thermal Engineering 27 (2007) 1883–1894 1891

Fig. 11. Temperature distributions (Pipe B) at 100, 500 and 1130 s.

Fig. 10. Temperature distribution (Pipe A) at 100, 500 and 1060 s.

ferred from the outside surface to the inside-surface.

5. Results and discussion This phenomenon is called the skin effect. After 300 s, the
differences of magnetic flux density and eddy current
In this study, a simulation of three different sizes of between the inside and the outside surfaces were less than
hollow steel rod for the induction heating process was per- that of the first 100 s. From Fig. 9b, it is clearly shown
formed. Fig. 6 shows the magnetic flux density and the eddy that the magnetic flux density and the eddy current on
current distribution on a cross-section of the workpiece both sides of the cross-section of the workpiece are
during different times of Pipe A. From Fig. 9a the magnetic smaller than other area. This is because that the relative
flux density on the area near the surface in the first 100 s is permeability (q) is decreasing as the temperature is increas-
higher than the inside area, therefore, the eddy current is ing. At times = 800 s, the magnetic flux density and the
concentrated on the outside-surface. The depth of the thin eddy current were found spread uniformly across in
contributed area is called the penetration depth [17]. Dur- the whole cross-section of the workpiece, as shown in
ing this time period, heat is generated in this area and trans- Fig. 8c.
1892 J.-Y. Jang, Y.-W. Chiu / Applied Thermal Engineering 27 (2007) 1883–1894

1200
1100
1000
900
800

Temperature (°C)
700
600
500
Pipe A
400 Experimental Outside T o
300 Experimental Inside T i
Numerical Outside T o
200
Numerical Inside T i
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. Sim-
ilarly, 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 propor-
tionally 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
800
Temperature (°C)

700
Fig. 10 shows the temperature distribution of Pipe A at
600
100, 500 and 1060 s, respectively. In Fig. 10a, owing to the
eddy current concentrated on both sides of workpiece near 500
the coil, the temperature on both sides is higher than that 400
Pipe B
of the central area. The maximum temperature is 363 C Experimental Outside To
300
and the minimum is about 200 C. In Fig. 10b, the temper- Experimental Inside Ti
ature distribution is the same as Fig. 10a. On account that 200 Numerical Outside To
the temperature reached the Curie point, the eddy current 100 Numerucal Inside Ti
was uniformly distributed in the workpiece and the heat 0
was transferred from the inside-surface to the outside-sur- 0 120 240 360 480 600 720 840 960 1080 1200
face, resulting in higher temperature on the inside-surface. Time (s)
The maximum temperature in Fig. 10b is 857 C and the Fig. 14. Experimental and numerical average temperature distribution for
minimum is about 787 C. In Fig. 10c, the maximum tem- Pipe B.
 J.-Y. Jang, Y.-W. Chiu / Applied Thermal Engineering 27 (2007) 1883–1894 1893

1100 surface temperature was higher than the outside-surface

1000
temperature. It is shown that the simulated temperature dis-
tributions are reasonable in agreement with the experimen-
900
tal data. For Pipe A, The maximum discrepancy between
800 numerical and experimental results for Pipes A, B and C
are 14.3%, 13.6% and 9.6%, respectively.
Temperature (°C)

700
The effect of the air gap between the coil and the work-
600
piece on the temperature distribution is shown in Fig. 16.
500 Table 3 shows 3 different air gaps in this study. It is found
400 that the temperature is higher when the air gap is smaller.
Pipe C A close look of this figure indicated that the average temper-
300 Exper imental Outside T o
Exper imental Inside T i
atures of the hollow steel for air gap = 5 mm are 10 C and
200
Numerical Outside To 15 C higher those for air gap = 15 mm, 25 mm, respec-
100 Numerical Inside T i tively. This is due to the fact that smaller air gap induces
0 stronger magnetic field throughout the workpiece.
0 120 240 360 480 600 720 840 960 1080 1200
Time (s) 6. Conclusion
Fig. 15. Experimental and numerical average temperature distribution for
Pipe C. The main purpose of this paper is to find what kind of
applied power distributions can result in an isothermal dis-
tribution between the inside and outside-surface of a hol-
850 low cylinder for the bonding process of a bi-metallic
tube. Three different sizes of the workpieces were numeri-
cally and experimentally investigated. The time variations
800 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
Temperature (°C)

750 the applied power input is found to be a steep-wise func-

Pipe A different gap tion (constant high power, 0–8 min, and decrease to it
Case 1 Outside Surface To
60%, 8–12 min, and then increase it original high power,
700 Case 1 Inside Surface Ti
12–20 min). The numerical results for the temperature dis-
Case 2 Outside Surface To tributions were reasonably in agreement with those of
Case 2 Inside Surface Ti experimental data. The maximum discrepancy for Pipes
650 Case 3 Outside Surface To A, B and C is 14.3%, 13.6% and 9.6%, respectively. It is
Case 3 Inside Surface Ti also shown that the temperature is increased as the air
gap is decreased. The average temperatures of the hollow
600 steel for air gap = 5 mm are 10 C and 15 C higher those
300 360 420 480 540 600 660 720 for air gap = 15 mm, 25 mm, respectively.
Times (s)

Fig. 16. Average temperature distributions of Pipe A for three different air Acknowledgement
gaps (from 300 to 720 s).

Financial support for this work was provided by the

China Steel Corporations, Taiwan, under contract CSC-
and the outside-surface of the workpiece. It should be noted
RE93603.
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 References
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