Study on the Uniformity of Temperature Distribution of Transverse Flux Induction Heating Based on a New Magnetic Pole

Abstract

A new magnetic pole is designed and proposed for the transverse flux induction heating (TFIH) device, since the TFIH device with the original magnetic pole has the deficiencies of uneven temperature distribution on the strip surface at the outlet of the heater and large magnetic resistance of the alternating magnetic field through the magnetic circuit. Using the magnetic–thermal coupling calculation method proposed in this paper, the eddy current field and temperature field of the TFIH device with the original and new magnetic poles are calculated and analyzed under the same excitation. At the same time, the temperature distribution on the strip surface of the TFIH device with the new magnetic pole is calculated under different excitation parameters, and the magnitude and frequency are obtained when the uniformity of the temperature distribution is best.

1. Introduction

Compared with the traditional open-fire heating method, the induction heating method has the advantages of high heating efficiency, energy saving and environmental protection, and it has become the preferred heating method in the heat treatment industry [1,2]. Based on Faraday’s electromagnetic induction law, the eddy current and moreover the eddy current loss can result on the workpiece when the alternating current is applied to its nearby winding, and thus the target workpiece will be heated, which can be called induction heating [3].

There are two basic induction heating methods: one is the longitudinal flux induction heating (LFIH) method and the other is the transverse flux induction heating (TFIH) method [4,5]. The magnetic induction lines are parallel to the workpiece in the LFIH, and thus it is only suitable for heating the workpiece with a large cross-sectional area, e.g., pipes and bars. The magnetic induction lines are perpendicular to the workpiece in TFIH, and thus it is suitable for heating the workpiece with a small cross-section area. For the heat treatment of metal strips, e.g., steel, the induction heating is a trend. The limitation of LFIH in heating strips is lower heating efficiency for the magnetic flux passes through a narrow area, i.e., strip cross section. Therefore, TFIH technology is widely used in strip heating industry [6]. Ref. [7] uses the Morris method of qualitative global sensitivity analysis (GSA) to rank the sensitivity values among design parameters and objectives of the TFIH device. Ref. [8] presents a TFIH system conformed by nonplanar inductors. These inductors are designed as nonplanar structures so that they can be assembled in overlapping structures, while preserving equivalence among them; in other words, they all present the same impedance. Ref. [9] considers the frequency-dependent inductive load and carries out the modeling process and transient performance analysis for the TFIH system. Ref. [10] investigates the electromagnetic forces generated by the TFIH device by using finite element numerical simulations. The results of four field calculation models are compared, and three methods were used to calculate the electromagnetic forces acting on nonmagnetic and magnetic thin plates. Ref. [11] presents a new structure of an induction heating device for aluminum alloy parallel hexahedral workpieces, where the device uses a magnetic field generated by a permanent magnetic inductor. Ref. [12] proposes an analytical model to evaluate the equivalent impedances of the TFIH system. The proposed analytical method can achieve good results in calculating the equivalent impedance of multiwindings with any number of windings in a TFIH system. In Ref. [13], Joule heat and temperature in conductive plate elements subjected to short-term induction heating by a nonstationary electromagnetic field are investigated. The research process of TFIH technology involves using existing or improved optimization algorithms (such as genetic algorithm, velocity-controlled particle swarm optimization algorithm, etc.) to optimize the design parameters of the TFIH device [14,15,16]. None of the above papers deal with the study of magnetic poles in TFIH.

Although the technology of TFIH has been greatly developed, it also has a disadvantage, that is, the uneven temperature distribution on the strip surface along the width direction of the heater outlet. This disadvantage seriously restricts the application and promotion of TFIH in the heat treatment industry.

In this paper, the current problems of TFIH technology are addressed, under the conditions of considering continuous strip movement and strip material parameters with temperature variation. The original magnetic pole is improved, and a new magnetic pole structure is proposed. At the same time, the temperature distribution on the strip surface of the heater with the new magnetic pole is calculated under different excitation parameters. It provides a theoretical basis for the actual heat treatment industry to make induction heaters and set appropriate excitation parameters.

2. Calculation Method of Magnetic–Thermal Coupling

MagNet, ThermNet and MATLAB were used to calculate and analyze the magnetic–thermal coupling. When the heating state is stable, the eddy current distribution on the strip surface under the heater does not change anymore, and the temperature distribution on the strip surface does not change without external disturbance. During the heating process, the surface temperature of the strip moving outside the heater outlet is the temperature at the heater outlet due to the continuous movement of the strip. Similarly, the strip surface temperature outside the heater inlet is the initial temperature of the strip, that is, the ambient temperature. In conclusion, only the eddy current and temperature distribution on the strip surface under the heater need to be studied.

Since the induction heater is symmetrical both transversely and longitudinally, only 1/4 of the model needs to be built. The area under the heater in the 1/4 model was meshed, and the calculation area of magnetic–thermal coupling is shown in Figure 1.

In Figure 1, X₁ represents the heater inlet; X₂₁ represents the heater outlet, and X₂ to X₂₀ represent the longitudinal grid dividing lines with equal spacing. Y₁ represents the edge position of the strip; Y₁₁ represents the midline position of the strip, and Y2 to Y10 represent the transversely spaced grid dividing lines. The distance between X_j and X_j+1 (j = 1, 2, …, 20) is 20 mm, which is represented by the letter d. The distance between Y_i and Y_{i + 1} (i = 1, 2, …, 10) is 30 mm. Each part of the region is represented by Q[i][j] (i = 1, 2, …, 10; j = 1, 2, …, 20).

In the calculation of magnetic–thermal coupling, the continuous movement of the strip is approximated to the stepping motion. Because the velocity of the strip motion (v) does not change, the calculation time of each step motion is t, which satisfies that

$$t=\frac{d}{v}$$

(1)

The calculation process of magnetic–thermal coupling is shown in Figure 2.

Before starting the calculation, the material parameters at various temperatures need to be assigned to the strip. At the beginning of the calculation, the first column of the strip is placed under the heater in the position Q[i][1] (i = 1, 2, …, 10). At this time, the initial temperature of the strip is 25 °C. The eddy current field and temperature field are coupled in time t. In Figure 2, the letter n represents the number of calculations of the magnetic–thermal coupling field during the software simulation calculation. After the termination time is reached, the temperature values in columns 1 to 19 of the strip are extracted, which is replaced by using the temperature replacement program of MATLAB; thus the material parameters of the strip change with the temperature. Then, the strip is moved forward by 20 mm. Column 1 and column 2 of the strip are at the Q[i][2], Q[i][1] (i = 1, 2, …, 10) positions under the heater, and the eddy current and temperature fields are coupled, and the material parameters are updated by the same method. When the strip is completely under the heater, the temperature values from column 2 to column 20 of the strip are extracted, and the temperature from columns 1 to 19 is assigned by using the temperature replacement program. At this time, the temperature values in columns 1 to 19 of the strip are changed into the temperature values in columns 2 to 20 in turn. Through the method of sampling point assignment, the continuous movement of the strip and change of the material parameters with temperature are realized.

It can be seen from many calculations and analyses that the trend of temperature stability on the strip surface under the heater is that the temperature gradually stabilizes from the inlet to the outlet of the heater. Therefore, by comparing the strip at the heater outlet, i.e., Q[i][20], the relative difference between the two average temperatures before and after the location is less than 1%, and the stability of the strip surface temperature under the heater can be determined. The calculation formula is as follows:

$$\frac{|T^{″}\mathrm{av}-T^{\prime }\mathrm{av}|}{T^{\prime }\mathrm{av}}\le 1\%$$

(2)

$$T^{″}\mathrm{av}=\frac{{\displaystyle \sum _{i=1}^{10}{T}^{″}\left[i\right]\left[\mathrm{20}\right]}}{10}$$

(3)

$$T^{\prime }\mathrm{av}=\frac{{\displaystyle \sum _{i=1}^{10}{T}^{\prime }\left[i\right]\left[\mathrm{20}\right]}}{10}$$

(4)

T′_av and T″_av are the average temperatures at the heater outlets calculated in the latter and the former times, respectively. T′[i][20] and T″[i][20] are the temperature values at each sample point on the strip surface at the outlet of the heater after the latter and the former calculations, respectively.

The calculation ends when the relative difference between the two is less than 1%. When the relative difference is more than 1%, the above method is continued to calculate the magnetic–thermal coupling, which realizes the continuous movement of the strip and material parameters changing with temperature.

When the heating state is stable, the relative nonuniformity of the surface temperature of the strip at the outlet of the heater is used to judge the quality of the heater and the heating effect, which is represented by the symbol T_rel. The calculation formula is as follows:

$$Trel=\frac{{\displaystyle \sum _{i=1}^{10}\left|T\right[i\left]\right[\mathrm{20}]-Tav|}}{10Tav}$$

(5)

T[i][20] is the temperature value of each sample point on the strip surface at the outlet of the heater after all the strip surface temperatures are stable. T_av is the average temperature of the strip surface at the outlet of the heater after all the strip surface temperatures are stable.

3. New Magnetic Pole Design

In this section, the original magnetic pole is improved, and a new magnetic pole structure is designed. The full model diagram and 1/4 model diagram of the induction heating model with the original magnetic pole are shown in Figure 3.

The heater model consists of magnetic pole, coil, strip and air bags. The magnetic pole material is an oriented silicon steel sheet with a thickness of 0.3 mm and a theoretical maximum core loss of 1.3 W/kg. A double-layer hexagonal coil was used in the coil structure as shown in Figure 4. The coil structure has a good heating effect, which makes the temperature distribution of the strip at the outlet more even. The coil material is made of copper, which has good electrical conductivity, corrosion resistance and processing performance.

The strip material is a high-quality carbon structural steel with a 0.42%~0.5% carbon content. Moreover, the material parameters of the strip change with the temperature, such as conductivity, magnetic conductivity, thermal conductivity and specific heat capacity.

The two-dimensional plan of the induction heater with the original magnetic pole is shown in Figure 5.

During the heating process, the magnetic field lines pass through the gap between the upper and lower magnetic poles of the heater twice, so the proportion of air in the magnetic circuit is larger. Because the magnetic permeability of the air is much smaller than that of the magnetic pole material, the effective value of the alternating magnetic field used to heat the strip in the heater with the original magnetic pole is small, which makes the heat source of the heating strip small and the temperature of the strip low. At the same time, through the simulation calculation and with the original magnetic pole, the temperature in the edge area of the strip is low, which is greatly different from the temperature in the middle area of the strip, resulting in a large nonuniformity of the surface temperature of the strip. In order to solve the problems existing in the heater with the original magnetic pole, a new structure of magnetic pole is proposed in this paper. The full model diagram and 1/4 model diagram of the induction heating model with the new magnetic pole are shown in Figure 6.

The original magnetic pole edge is flush with the strip edge. The new magnetic pole is based on the original magnetic pole, which is lengthened along the width direction of the strip as a whole, and the edge part of the lengthened is connected. At the same time, the air gap between the original magnetic pole and the strip is uniform, and the air gap between the new magnetic pole and the strip is uneven. The gap between the new magnetic pole and the strip increases from the edge area of the strip to the midline area of the strip, which is a “stepped” manner.

The two-dimensional plan of the induction heater with the new magnetic pole is shown in Figure 7. Because the magnetic flux will flow along the path with small magnetic resistance in the magnetic circuit [17], the magnetic flux generated by the heater coil passes through the air gap between the pole and the strip only once in the process of flow through the new magnetic pole, so the proportion of air in the magnetic circuit decreases, and the magnetic resistance of the alternating magnetic field through the magnetic circuit decreases.

In conclusion, the effective value of the alternating magnetic field used to heat the strip in the induction heater with the new magnetic pole is large, which makes the heat source of the heating strip large and the temperature of the strip high. At the same time, the air gap between the new magnetic pole and the strip is uneven, which can improve the surface temperature distribution of the strip.

4. Results

For the design improvement of the original magnetic pole structure, the parameters set by the original magnetic pole heater 1/4 model are shown in

Table 1. The parameters of 1/4 model of TFIH with the original magnetic pole.

Parameters	Values
Current magnitude/A	1100
Current frequency/Hz	500
Speed of strip movement/(m/s)	0.1
Air gap between strip and magnetic pole/mm	3
Coil cross-section area/mm × mm	28 × 14
Magnetic pole size/mm × mm × mm	400 × 300 × 60
Strip size/mm × mm × mm	400 × 300 × 0.25
Air bag size/mm × mm × mm	4000 × 600 × 123.25

.

The heater model with the new magnetic pole is based on the parameters shown in Table 1 with the following main differences:

• The width of the new magnetic pole is 320 mm;
• The air gap between the new magnetic pole and the strip is shown in

  Table 2. The air gap between the new magnetic pole and the strip.

  Distance from the Edge of the Strip/mm	Air Gap between the New Magnetic Pole and the Strip/mm
  0–100	2.5
  100–200	2.75
  200–300	3

  .

Using the magnetic–thermal coupling calculation method proposed in this paper, and under the condition that the material parameters of the strip change with temperature and the continuous movement of the strip during heating process, the 1/4 models of the induction heater with the original and new magnetic pole were calculated and analyzed under the same excitation. The results are as follows:

When the heating state is stable, the magnetic flux density distribution in the original and new magnetic pole models is shown in Figure 8 and Figure 9.

Since the magnetic flux density distribution cannot be clearly displayed in the 1/4 model, the 1/2 model was selected. As shown in Figure 8, in the original magnetic pole model, air accounts for a large proportion of the magnetic circuit, which results in a large magnetic resistance and a small magnetic flux density in the original magnetic pole, resulting in a small effective value of the alternating magnetic field used for the heating strip. As shown in Figure 9, in the new magnetic pole model, the magnetic field lines generated by the heater coil pass through the new magnetic pole, and the proportion of air in the magnetic circuit is small, which makes the magnetic resistance small. The magnetic flux density in the new magnetic pole is significantly higher than that in the original magnetic pole, resulting in the effective value of the alternating magnetic field used for the heating strip being large. At the same time, the new magnetic pole model improves the mutual offsetting of the magnetic flux density between the two heating coils.

When the heating state is stable, the eddy current distribution on the strip surfaces of the two models is shown in Figure 10.

In Figure 10, RMS |J| represents the root mean square value of the eddy current on the strip surface. As shown in Figure 10, the eddy current distribution on the strip surfaces of the two models follows ‘the law of coil projection’, whereby the eddy current distribution on the strip surface is concentrated in the positive projection area of the heater coil on the strip surface. The eddy current distribution of the two models also satisfies that the eddy current value of the strip surface at the outlet of the heater is lower than that at the inlet of the heater. During the heating process, the strip moves continuously, and the strip moves from the heater inlet to the heater outlet, and the surface temperature of the strip rises continuously. The magnetic conductivity and electrical conductivity of the strip decrease with increasing temperature. The eddy current value on the strip surface is positively correlated with the magnetic conductivity and electrical conductivity of the strip. Therefore, the eddy current value on the strip surface at the outlet of the heater is lower than that at the inlet of the heater.

As can be seen from Figure 10, the strip surface eddy current value of the new magnetic pole model is generally larger than that of the original magnetic pole model. Because the effective value of the alternating magnetic field used for the heating strip is large in the new magnetic pole model, the induced voltage on the surface of the strip is large, thus the eddy current on the surface of the strip is large.

When the heating state is stable, the heat source distribution on the strip surface of the two models is shown in Figure 11.

As shown in Figure 11, the heat source distribution on the strip surface of the two models is similar to that of the eddy current distribution, and they both follow the law of coil projection. It is also observed that the distribution of heat sources on the surface of the strips of the two models satisfies that the value of heat sources at the outlet of the heater is larger than that at the inlet of the heater. The temperature of the strip surface and the heat source affect each other during the heating process, and the surface temperature of the strip increases from the heater inlet to the heater outlet, and the electrical conductivity of the strip decreases with the increase in the temperature, and the degree of decrease with the increase in temperature is greater than that of the eddy current value on the strip surface. The relationship between the Joule thermal power per unit volume of heated strip and the surface eddy current and electrical conductivity of the strip is

$$Pv=\frac{|J{|}^2}{\sigma }$$

(6)

where |J| is the eddy current (A/m²), and σ is the electrical conductivity. Therefore, the heat source value of the strip surface at the heater outlet is greater than that at the heater inlet.

When the heating state is stable, the temperature distribution on the strip surface of the two models is shown in Figure 12, and the temperature distribution on the strip surface at the heater outlet of the two models is shown in Figure 13.

As shown in Figure 12 and Figure 13, the temperature on the strip surface of the two models increases continuously with the movement of the strip from the heater inlet to the heater outlet. Meanwhile, the temperature distribution of each part of the strip surface is smooth due to the heat transfer. Due to the convection and radiation heat dissipation between the strip edge area and air, the temperature of the strip edge area is lower than that of the middle area, resulting in poor uniformity of the temperature distribution of the original magnetic pole model. Due to the multiple effects of “edge effect of electromagnetic field” and new magnetic pole structure, the temperature in the edge area of the strip is close to that in the middle area of the strip, which makes the temperature distribution at the outlet of the heater more uniform in the new magnetic pole model.

When the strip surface temperature reaches a steady state, the parameters to evaluate the performance of the transverse flux induction heating can be obtained: maximum temperature of the strip surface at the heater outlet T_max, average temperature of the strip surface at the heater outlet T_av and relative nonuniformity of the temperature distribution on the strip surface at the heater outlet T_rel.

From

Table 3. The evaluation parameters of TFIH.

	Tmax/°C	Tav/°C	Trel/%
The original magnetic pole model	1079.4	919.2	11.53
The new magnetic pole model	1289.8	1244.5	2.78

, it can be seen that the new magnetic pole model T_max is 210.4 ℃ higher than the original magnetic pole model, and T_av is 325.3 ℃ higher than the original magnetic pole model. The reason is that the eddy current value and heat source value of the new magnetic pole model strip surface are larger than the original magnetic pole model overall, so the temperature value of the strip surface is also larger than the original magnetic pole model overall. At the same time, the new magnetic pole model T_rel is 8.75% smaller than the original one, and the temperature distribution on the strip surface at the outlet of the heater with the new magnetic pole is better.

In summary, to overcome the deficiencies of the uneven temperature distribution on the strip surface at the outlet of the heater with the original magnetic pole and the large magnetic resistance of the origin magnetic pole, a new magnetic pole for TFIH was designed based on the original magnetic pole. MagNet, ThermNet and MATLAB were used to calculate and analyze the magnetic–thermal coupling. The results of the calculations and analysis show that under the same excitation conditions, the heater with the new magnetic pole can heat the strip to a higher temperature, and the uniformity of the temperature distribution on the surface of the strip is better, which improves the heating effect. In practical engineering applications, if the strip needs to be heated to a specific temperature, changing the excitation conditions of the heater can meet the requirements. The TFIH device with the new magnetic pole presented in this paper provides a theoretical basis for the manufacture of heaters in practical projects and has a high application value.

5. Effect of Excitation Parameters on Strip Surface Temperature Distribution

5.1. Effect of Current Magnitude on Strip Surface Temperature Distribution

The current frequency is 500 Hz, and the speed of the strip movement is 0.1 m/s when different currents are applied to the coils of the heater with the new magnetic pole. The strip surface temperature distribution was calculated and compared for the excitation currents of 600 A, 800 A, 1000 A, 1100 A and 1200 A. The evaluation parameters of TFIH for different excitation currents are shown in

Table 4. The evaluation parameters of TFIH for different excitation currents.

Current Magnitude/A	Tmax/°C	Tav/°C	Trel/%
600	287.5	269.6	3.93
800	550.9	526.9	2.77
1000	957.8	920.8	2.50
1100	1289.8	1244.5	2.78
1200	1592.9	1521.0	4.09

. The temperature distribution on the strip surface at the heater outlet for different excitation currents is shown in Figure 14.

It can be seen from Table 4 and Figure 14 that T_max and T_av increase, and T_rel tends to decrease and then increase as the excitation current magnitude increases. At the same time, according to the software calculation process, the number of calculations of the coupling field remains the same when the strip surface temperature is stable with the increase in the excitation current magnitude. Based on the above data and their distribution law, the current magnitude was set to 1000 A in Section 5.2 to study the effect of the excitation current frequency on the strip surface temperature distribution under the principle of T_rel minimization.

5.2. Effect of Current Frequency on Strip Surface Temperature Distribution

In the study of the effect of the excitation current frequency on the strip surface temperature distribution, according to the calculation results in Section 5.1, the current magnitude was set to 1000 A and the strip movement speed was 0.1 m/s. The strip surface temperature distribution was calculated and compared for the excitation current frequencies of 100 Hz, 300 Hz, 500 Hz, 700 Hz and 900 Hz, and the evaluation parameters of TFIH for different excitation current frequencies are shown in

Table 5. The evaluation parameters of TFIH for different excitation current frequencies.

Current Frequency/Hz	Tmax/°C	Tav/°C	Trel/%
100	559.5	504.3	5.92
300	769.7	751.0	1.44
500	957.8	920.8	2.50
700	1116.7	1057.0	3.05
900	1225.6	1162.3	3.49

. The temperature distribution of the strip surface at the heater outlet for different excitation current frequencies is shown in Figure 15.

From Table 5 and Figure 15, when the excitation current frequency increases, T_max and T_av increase, and T_rel shows a trend of decreasing and then increasing. At the same time, according to the software calculation process, the number of calculations of the coupling field remains the same when the strip surface temperature is stable with the increase in the excitation current frequency.

Based on the above data and their distribution law, the optimal combination of excitation parameters is 1000 A and 300 Hz, as T_rel is the smallest when the frequency is 300 Hz, based on the principle of optimal uniformity of the strip surface temperature distribution.

6. The Limitations of the Model

Compared with the original magnetic pole model, the model with the new magnetic pole has a certain degree of difficulty in processing and manufacturing. Since the new magnetic pole structure and the air gap between the new magnetic pole and the strip are not uniform, this makes the processing and manufacturing process more demanding in terms of processing accuracy. But the effective value of the alternating magnetic field of the model with the new magnetic pole for heating the strip is larger, and the average temperature of the strip surface at the heater outlet and the uniformity of the temperature distribution are better than those of the model with the original magnetic pole. Although the new magnetic pole will make the processing and manufacturing more accurate, the performance of the heater with the new magnetic pole is better than the model with the original magnetic pole.

7. Conclusions

To overcome the deficiencies of uneven temperature distribution on the strip surface at the outlet of TFIH with the original magnetic pole and the large magnetic resistance of the alternating magnetic field through the magnetic circuit, the original magnetic pole is improved, and a new magnetic pole is proposed. Meanwhile, in the new magnetic pole model, the distribution of the strip surface temperature under different excitation parameters was analyzed, and a best set of excitation current magnitude and frequency was obtained. The main conclusions are as follows

• The effective value of the alternating magnetic field of the heater with the new magnetic pole for heating the strip is larger.
• The induction heater with the new magnetic pole can produce a higher temperature, and the uniformity of the temperature distribution is better than that of the heater with the original magnetic pole.
• Under the combination of excitation parameters with an excitation current magnitude of 1000 A and a frequency of 300 Hz, the uniformity of the temperature distribution on the strip surface at the heater outlet in the direction of the strip width is the best, and its relative temperature nonuniformity is 1.44%, which can effectively meet the needs of actual industrial production.
• With the increase in the excitation current and frequency, the maximum temperature T_high and the average temperature T_av of the strip surface at the heater outlet increase.
• The change of excitation current magnitude and frequency does not affect the number of calculations of the coupling field when the strip surface temperature is stable.