Issue 
Metall. Res. Technol.
Volume 119, Number 1, 2022



Article Number  103  
Number of page(s)  17  
DOI  https://doi.org/10.1051/metal/2021098  
Published online  20 December 2021 
Regular Article
Research on an external adjustment method for RPECT roll profiles based on the segmented cooling principle
^{1}
National Engineering Research Center for Equipment and Technology of Cold Strip Rolling, Yanshan University, Qinhuangdao 066004, Hebei, China
^{2}
College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei 066004, China
^{*} email: whj2011@ysu.edu.cn
Received:
24
September
2021
Accepted:
29
November
2021
Roll profile electromagnetic control technology (RPECT) is a new strip flatness control technology that changes roll gap shape by controlling the roll profiles of electromagnetic control rolls (ECRs). To address the randomness of the flatness defect locations, this paper proposes an external adjustment method for RPECT roll profiles based on the segmented cooling principle. Based on the layout of the cooling areas and electromagnetic sticks, an electromagneticthermalstructural coupled model is established to analyse roll profile variations. The results show that symmetrically changing the cooling intensities of the different cooling areas can increase or decrease the roll crown of the ECR, while asymmetrically changing the cooling intensities of the different cooling areas can change the position of the maximum bulging point of the ECR. Variations in the component cooling ratio coefficient impact the effects of different cooling strategies, which needs to be considered when selecting the cooling strategy configuration scheme. Compared the maximum bulging values, radial temperature gradients and axial temperature gradients of different electromagnetic stick (ES) structures, the regulation law reverses when the length of the ES is too small, and the variation of the law is very small. Therefore, different ES structures have different segmented cooling regulation characteristics.
Key words: roll profile electromagnetic control technology / segmented cooling / component cooling ratio coefficient / electromagnetic stick structure
© EDP Sciences, 2021
1 Introduction
The roll gap shape is the main control index of the strip rolling process and can be used to control strip thickness and flatness. Roll profile electromagnetic control technology (RPECT) is a new roll gap shape control technology with multistage roll profile regulation and flexible roll crown control. In RPECT, an electromagnetic stick (ES) installed in the roll’s inner hole is selected as the basic unit element and then induction heated in the regulation process. Through the thermal expansion of the ES and the internal constraint of the roll, the contact force between the ES and the roll can be generated. The contact force and the internal heat of the roll form the thermal force bulging of the electromagnetic control roll (ECR). Due to the thermalforce bulging characteristic, the electromagnetic control roll profile has certain temperature sensitivity, so it is suitable for cold rolling with weak heat exchange between strip and roll. Considering that the technology has flexible roll profile control ability, it can be loaded on the working roll of the cold rolling mill. Regarding RPECT, Liu et al. [1,2] established an electromagneticthermalstructure coupled model to analyse the problem of roll profile control in RPECT and proposed a roll profile test technology to conduct experimental research on RPECT. Feng et al. [3] analysed the ability of a largesize sleeved backup roll to control its roll profile. Yang et al. [4] studied the electromagnetic control characteristics of RPECT within the ES limit control range. The above research provides support for the development of RPECT, proving that RPECT has a certain roll profile control ability. However, the position of the ES in the roll cannot be changed during the RPECT process. When the flatness defect position is not entirely within the influence area of the electromagnetic control roll profile, changing the electromagnetic parameter values of different ES induction coils can change the position of the influence area, but the position adjustment ability is limited. Therefore, it is necessary to consider using external control methods to change the influence area of the electromagnetic control roll profile to address the problem of flatness defects.
According to Liu et al. [5], roll profiles can be changed by using roll surface cooling for ECRs. Therefore, the liquid cooling method can be considered an external method of adjusting the roll profile in RPECT. The liquid surface cooling method, which is based on convective heat transfer, has been widely studied and applied. Horský et al. [6] studied the effect of spray cooling intensity on the final material structure in the hot rolling process of long products. Wang et al. [7] developed a twodimensional finite difference program based on an inverse heat conduction model that calculated the local surface convective heat transfer coefficients and analysed the effect of the cooling water jet flow rate on the surface temperature. Labergue et al. [8] investigated the cooling of hot surfaces in full cone sprays and compared the results with the cooling effects of liquid jets. Smakulski et al. [9] analysed the heat removal capabilities of direct cooling technology by using porous media, microchannel heat sinks and spray cooling. Saboonchi et al. [10] studied the effect of water spray geometric parameters on the working roll temperature. The above research focused on the heat transfer effect of overall cooling for different objects. However, there have been few studies on the impact of zoning and subsection cooling. In the field of RPECT, the effects of zoning and subsection cooling have not been studied.
To improve the flexibility of RPECT, this paper proposes using external segmented cooling to control the heat axial transfer in ECRs and the roll profile. Given the importance of the roll profile segmentation control strategy on the roll profile status, this paper compared and studied the bulging ability, crown control ability, ECR temperature field variation and adaptability of different ES structures, as well as discussed the advantages and disadvantages of different roll profile segmentation control strategies.
2 Principle of the external adjustment method of RPECT roll profiles and model establishment
2.1 Principle of the external adjustment method of RPECT roll profiles
The RPECT roll profile can be used as a kind of displacement roll profile to participate in the roll gap shape control process. In the flatness control process, the roll profile of ECR can be changed to another roll profile according to the demand of the strip control. The RPECT roll profile consists of a force contribution roll profile and a thermal contribution roll profile. The thermal contribution roll profile exists because heat from the ES after electromagnetic induction heating can be transferred to the ECR, causing the ECR to thermally bulge. The force contribution roll profile is due to the contact area of the ES being heated, causing the ECR to bulge. In the general control, the rapid variation of the roll profile relies on the electromagnetic parameters and the control temperature of the ES, and the buling amount of 160 µm can be realized in 300 s. If the demands for the regulation time and the bulging amount increase, the magnetic parameter value and the ES control temperature value are often increased. However, the heat source is located in the induction heating zone of the ES, the response time of the force contribution roll profile is less than that of the thermal contribution roll profile in the control process. Therefore, on the basis of weakening the thermal contribution roll profile, the control mode based on force contribution roll profile is conducive to shorten the time to reach the target roll profile and improve the stability of the target roll profile. According to the results in the literature [5], the surface cooling strategy mainly affects the thermal contribution roll profile while having little effect on the force contribution roll profile. This paper proposes an external adjustment method for RPECT roll profiles, which can improve the flexibility of RPECT. In this control method, the temperature field of different ECR segments can be changed by adjusting the external cooling intensities of different segments and changing the thermal contribution roll profile. Figure 1 shows a diagrammatic sketch of the external adjustment method of the RPECT roll profile.
The segmented electromagnetic stick can be thought of as a basic component for adjusting the RPECT roll profile. The external cooling area of the segmented electromagnetic stick is divided into three areas: cooling area I, cooling area II and cooling area III. Cooling area I and cooling area III are symmetrical and equally spaced on both sides of cooling area II. According to different control strategies, changing the cooling intensities of different cooling areas can change the temperature field of the ECR to achieve the adjustment of the comprehensive roll profile.
Fig. 1 Diagrammatic sketch of the division method of cooling areas of the surface of the ECR. 
2.2 Model establishment
The RPECT process involves the simulation of the electromagnetic field, temperature field and stress field, which is an electromagneticthermalstructural coupling problem. Therefore, the electromagneticthermalstructure module of MARC software must be used to analyse this process. Due to the large simulation requirements of the threedimensional multifield coupling model, the RPECT FE model needs to be simplified. The ECR, ES and induction coils are all revolving bodies with the same revolving axes, and the model can be simplified as an axisymmetric model to reduce the calculation time. In the process of roll cooling, the heat transferred to the rolls is decreased with time, and the distribution of circumferential cooling zone is periodic. Cerni et al. [11] converts convective cooling conditions along the circumference into a linear heat source. Troeder et al. [12] uses the same assumption to analyse the transient stress of hot rolled shell. Therefore, this FE model assumes that the external cooling is a linear heat source, and simplifies the model to an axisymmetric model, which is convenient for simulation. Based on the principles of the external adjustment method for the RPECT roll profile, variations in the ECR profile under different external adjustment methods can be simulated by changing the cooling intensity of the roll surface and the section lengths of the same cooling intensities. The configurations of the ES structure and cooling areas are the same as those in Figure 1. The induction zone temperature control mode is used as the control mode. In this mode, the temperature of the induction heating zone can be controlled by dynamically adjusting the induction current to ensure that the temperature of the induction zone is essentially stable at the target temperature. Table 1 shows the parameters of the FE model.
Based on the above parameters, the RPECT electromagneticthermalstructural axisymmetric model is established in Figure 2. To ensure the simulation accuracy of the electromagnetic field and temperature field, the model includes not only the ECR, ES and induction coil, but also air units. The outermost cells of the air units are set as potential zero and magnetic potential zero. Contact heat transfer occurs between the ECR and the ES. According to Yovanovich et al. [13], the contact heat transfer ability is related to the contact stress and contact face. Because hot charging is used to assemble the ECR and the ES, prestress exists in the contact area of RPECT. Therefore, the ECR and ES must be in good contact before regulation. According to the literature [5], the heat transfer coefficient is a constant with a value of 3 kW/(m^{2 }· K). The heat transfer between the RPECT element and air is comprehensive heat transfer include convective heat transfer and radiative heat transfer, and the coefficient is 0.03 kW/(m^{2 }· K) [14]. The initial temperature of the model is 30 °C.
Parameters of the FE model.
Fig. 2 RPECT electromagneticthermalstructural axisymmetric model with segmented cooling. 
2.3 Model validation
To verify the accuracy of this model, the roll profile electromagnetic control experiment platform is used as shown in Figure 3a. The platform includes: a power supply, an ECR, a temperature detector, an acquisition system, acquisition channels and a test system. According to the data of the acquisition channels, the radial bulging of the roll can be calculated by Formula (1).(1) (2)where θ is the angle of a foil gauge, R is the original radius of the roll, l is the original length of a foil gauge, ԑ_{c} is the strain value of a strain gauge, Δθ is the angle variation after bulging, ΔR is the radius variation after bulging.
In RPECT, the bulging of roll is circumferential uniform bulging. The Δθ is zero, ΔR is equal to ԑ_{r}R. ԑ_{r} is the radial strain value of the roll. Therefore, Formula (1) and (2) can be combined into Formula (3).(3)
To sum up, Formula (3) can be used to calculated the radial bulging. The experimental conditions were as follows: the structure parameters of ECR and ES are the same as Table 1, the initial temperature of the model is 17 °C, the heating model of ES is continue heating, and the cooling mode of roll surface is air cooling with the cooling intensity is 0.03 kW/(m^{2 }· K). The control current frequency is 400 Hz, and the variation of the current is shown in Figure 3b.
Figure 4 is the temperature variation of ES induction area and the roll crown variation of ECR. The results show that the maximum temperature difference between experiments and simulations is less than 8.4 °C, and the roll crown difference is less than 2.2 µm. The experimental and simulation error of ES temperature is 5.33%, and the experimental and simulation error of roll crown is 6.36%. The experimental results and the simulation results show the FE model has a high accuracy, and can be used to analyse the control ability of RPECT.
Fig. 3 Roll profile electromagnetic control experiment platform and the variation of the current. (a) roll profile electromagnetic control experiment platform and (b) the variation of the current. 
Fig. 4 Temperature variation of ES induction area and the roll crown variation of ECR with time. (a) temperature variation, and (b) roll crown variation. 
3 Results and discussion
3.1 Analysis of the symmetrical segmented cooling strategy
To increase and decrease the roll profile control ability of RPECT, a symmetrical segmented cooling strategy is proposed. In this strategy, cooling areas I and III are considered the edge cooling areas, while cooling area II is considered the middle cooling area. The cooling intensity of the edge cooling area and middle cooling area can be adjusted to change the roll profile control ability of RPECT. The cooling intensities of cooling areas I and III are equal, with the heat transfer coefficient of HTCE, while the heat transfer coefficient of the middle cooling area is HTCM. The length of the cooling area is 100 mm, and the cooling intensity ranges from 0.03 kW/(m^{2 }· K) to 3 kW/(m^{2 }· K). Based on the parameter differences of the cooling areas, the symmetrical segmented cooling strategy can be divided into Strategy I, which keeps HTCM constant while changing HTCE, and Strategy II, which changes HTCM while keeping HTCE constant.
3.1.1 The bulging ability and the variation of the roll crown
To analyse the effects of the symmetrical segmented cooling characteristics, the bulging amount of the edge cooling area is defined as C1, the bulging amount in the middle cooling area is defined as C2, the bulging amount of the roll edge is defined as C3, and the roll crown is defined as C, as shown in Figure 1. The roll crown is the difference between C2 and C3.
Figure 5 shows the variations of C1, C2 and C for different HTCM values when HTCE is changed. When HTCM is kept constant while HTCE is increased, C1 and C2 gradually decrease while C gradually increases. The reason for this is that increasing HTCE can improve heat exchange at the edge of the ECR and reduce bulging at the edge of the ECR. Due to the whole temperature field of the ECR, temperature variations at the edge of the ECR affect the temperature variations in the middle of the ECR. Therefore, changing HTCE can affect the values of C1 and C2. As HTCM is increased from 0.03 kW/(m^{2 }· K) to 3 kW/(m^{2 }· K), the variations of C1 with increasing HTCE are 6.17 µm, 6.01 µm, 5.8 µm, 5.57 µm; the variations of C2 with increasing HTCE are 3.02 µm, 2.25 µm, 1.69 µm, 1.43 µm; and the variations of C with increasing HTCE are 0.06 µm, 1.22 µm, 1.74 µm, 1.99 µm. For different HTCMs, the effects of increasing HTCE are the same for C1 and C2. When the cooling intensity of the middle cooling area is high, the effect of increasing HTCE on C1 and C2 is weakened. The reason for this is that the ES is the thermalforce bulging driving device of RPECT, and its installation position is the same as that of the middle cooling area. When the cooling intensity of the middle cooling area is increased, the heat inside the ECR is reduced, and thus the effect of increasing HTCE on the roll profile of RPECT is reduced. Therefore, strategy I can improve the roll crown of the ECR, but the variation in the ECR roll crown is small.
Figure 6 shows the variations in C1, C2 and C for different HTCE values while HTCM is changed. When HTCE is kept while HTCM is increased, then C1, C2 and C decrease. The reason for this is that the installation position of the ES is the same as that of the middle cooling area, and thus increasing HTCM can increase the radial heat transfer ability, decreasing the whole temperature of the ECR. This phenomenon leads to a decrease in the overall bulging amount of the ECR. As HTCE are increased from 0.03 kW/(m^{2 }· K) to 3 kW/(m^{2 }· K), the variations of C1 with increasing HTCM are 1.64 µm, 2.01 µm, 2.03 µm, 2.47 µm; the variations of C2 with increasing HTCM are 13.07 µm, 11.87 µm, 11.69 µm, 11.48 µm; and the variations of C with increasing HTCM are 13.59 µm, 12.48 µm, 11.96 µm, 11.66 µm. For different HTCE, the effects of increasing HTCM are the same for C1 and C2, and the trend can be described as follows: as HTCE increases, the variation decreases. The impact on C1 is less than on C2. As a result of strategy II, the roll crown can be decreased with increasing HTCM. The difference between the cases is that as HTCM increases, the bulging amount of the ECR edge decreases, while the bulging amount of the ECR middle remains relatively constant. Because the variation of the bulging amount at the ECR edge is smaller than that in the ECR middle, the effects of roll crown weakening are nearly identical under different cases of strategy II. Therefore, strategy II has the effect of reducing the roll crown of the ECR. A comprehensive comparison of the results in Figure 5 and Figure 6 shows that the maximum effect of roll crown weakening can reach 13.59 µm, while the maximum effect of roll crown enhancement is 2 µm. The roll crown weakening ability is stronger than the roll crown enhancement under different cooling strategies.
Fig. 5 Variations in C1, C2 and C under Strategy I. (a) C1, (b) C2 and (c) C. 
Fig. 6 Variations of C1, C2 and C under Strategy II. (a) C1, (b) C2 and (c) C. 
3.1.2 The ECR temperature gradient
To further evaluate the state of the thermal contribution roll profile under Strategies I and II, P1, P2 and P3 are selected as marked points, as shown in Figure 7. P1 is the midpoint of the roll surface; P2 is the midpoint of the radial line in the middle of the ECR, 45 mm away from P1; and P3 is at the edge of the ES effect area, on the ring surface of the ECR with the same radius as P2. Based on the above points, radial and axial temperature gradients can be determined.
The radial temperature gradient is the temperature gradient from P2 to P1, denoted as GradT_{r}. The axial temperature gradient is the temperature gradient from P2 to P3, denoted as GradT_{a}. The calculation formulas are as follows:(4) (5)where T_{P1} is the temperature of P1, T_{P2} is the temperature of P2, T_{P3} is the temperature of P3, and l_{ES} is the length of the ES.
Figure 8 shows the variations in the radial and axial temperature gradients under Strategy I. In Figure 8, when HTCM is increased from 0.03 kW/(m^{2 }· K) to 3 kW/(m^{2 }· K), the variations in GradT_{r} with increasing HTCE are −0.0032 °C/mm, 0.0132 °C/mm, −0.008 °C/mm, and −0.0095 °C/mm; and the variations in GradT_{a} with increasing HTCE are 0.0345 °C/mm, 0.0363 °C/mm, 0.0362 °C/mm, and 0.0359 °C/mm. Based on the contact relationship between the ES and the ECR, the inner section of the ECR can be divided in two sections: the roll section with the ES and the roll section without the ES. As HTCE increases, the temperature of the roll section without the ES decreases. Marking point P3 is located at the intersection of the roll section with the ES and the roll section without the ES. Under Strategy I, the temperature of P3 can be decreased, leading to an increase in GradT_{a}. Marking points P1 and P2 are located in the middle of the roll section with the ES. Although changing HTCE can affect the temperature of this section, the effect is not obvious, and the temperature variations of P1 and P2 are small. Therefore, GradT_{r} is essentially constant.
Figure 9 shows the variations in the radial and axial temperature gradients under Strategy II. The results show that with increasing HTCM, GradT_{r} increases and GradT_{a} decreases. In Figure 9, when HTCE is increased from 0.03 kW/(m^{2 }· K) to 3 kW/(m^{2 }· K), the variations in GradT_{r} with increasing HTCM are 0.2410 °C/mm, 0.2272 °C/mm, 0.2219 °C/mm, and 0.2183 °C/mm; and the variations in GradT_{a} with increasing HTCM are −0.0478 °C/mm, −0.0478 °C/mm, −0.0474 °C/mm, and −0.0469 °C/mm. Therefore, when HTCE is different, the difference in GradT_{a} is small, and changing HTCM can affect the temperature of the roll section with the ES, increasing the temperature difference between P1 and P2. The reason for this is that the temperature increase in the roll section without the ES is due to heat transfer from the roll section with the ES, and the cooling effect of the middle cooling area can affect the temperature distribution in the roll section without the ES.
The above results show that in the symmetrical segmented cooling strategy, changing the cooling intensity of different cooling areas symmetrically can enhance or weaken the ECR roll crown. The reason for the crown variation is that the use of symmetrical segmented cooling can change the heat contribution roll profile, which leads to directional changes in the comprehensive roll profile.
Fig. 7 Schematic diagram of the temperature gradient marking points. 
Fig. 8 Variations of the radial and axial temperature gradients under Strategy I. (a) GradT_{r} and (b) GradT_{a}. 
Fig. 9 Variations of the radial and axial temperature gradients under Strategy II. (a) GradT_{r} and (b) GradT_{a}. 
3.2 Analysis of symmetrical segmented cooling characteristics with variable coefficients
To analyse the effect of the cooling area length, this section analyses the roll profile control ability under different cooling area length ratios. A component cooling ratio coefficient that can quantify the configuration of symmetrical segmented cooling is proposed. The coefficient is defined as the ratio of the length of cooling area II to the length of the ES; the formula is shown in Formula (6).(6)where η is the component cooling ratio coefficient, l_{CII} is the length of cooling area II, and l_{ES} is the length of the ES.
The basic length of the ES is 150 mm, and the basic cooling intensity is 0.03 kW/(m^{2 }· K). The cooling intensity ranges from 0.03 kW/(m^{2 }· K) to 3 kW/(m^{2 }· K).
3.2.1 The bulging ability and the variation of roll crown
Figure 10 shows the variations in C1, C2 and C under Strategy I for different η. When comparing the results to those in Figure 5, changing η does not affect the variation trend of C1 and C2, but it can affect the variation of C. With increasing η, the variations of C1 are −8.27 µm, −6.01 µm, and −5.28 µm; the variations of C2 are −7.39 µm, −3.02 µm, and −1.21 µm; and the variations of C are −4.14 µm, 0.06 µm, and 2.34 µm. When η is increased, the effect of increasing HTCE on the control ability of C1 and C2 is weakened. The difference in the weakening effect is that the effect on C1 is weakened as a whole, whereas the effect on C2 is decreased slowly when the cooling intensity is 0.03 kW/(m^{2 }· K). Due to the difference between C1 and C2, C can be changed from a decreasing trend to an increasing trend with increasing HTCE. The reason for this is that when the value of η is small, the edge cooling areas have a greater effect on the temperature field in the middle of the ECR and can increase the variations of C2, resulting in the transformation from crown enhancement to crown weakening. When the value of η is large, the temperature field in the middle of the ECR is less likely to be affected by the edge cooling areas. With increasing η, the variation in the C2 value decreases, and the value of C1 decreases, thus ensuring that the crown is enhanced. Therefore, when configuring the symmetrical segmented cooling strategy in RPECT for crown enhancement, it is necessary to take into account the length distribution of the cooling areas. If η is too small, it is difficult to achieve the effect of roll crown enhancement, but it is easy to produce the effect of crown weakening.
Figure 11 shows the variations of C1, C2 and C under Strategy II for different η. When the value of η is changed, the variation trends of C1, C2 and C are the same as those in Figure 6. Increasing η decreases the control ability of C1 while increasing the effects of changing HTCM on C2 and C. Therefore, increasing η can increase the crown weakening ability.
Fig. 10 Variations in the values of C1, C2 and C under Strategy I for different η. (a) C1, (b) C2, and (c) C. 
Fig. 11 Variations in the values of C1, C2 and C under Strategy II for different η. (a) C1, (b) C2, and (c) C. 
3.2.2 The ECR temperature gradient
Figure 12 shows the variations in GradT_{r} and GradT_{a} under Strategy I for different η. The results show that for different η, the trends of GradT_{r} and GradT_{a} with increasing HTCE do not change. With increasing η, the variations in GradT_{r} are 0.0484 °C/mm, 0.0132 °C/mm, and 0.0037 °C/mm, and the variations in GradT_{a} are 0.0069 °C/mm, 0.0207 °C/mm, and 0.0187 °C/mm. The increase in η decreases the effect of HTCE on GradT_{r}, and its effect on GradT_{a} can be described as follows: the effect on GradT_{a} is first increased to a certain value and then decreased. In Figure 12b, the effects of the cases when η=0.67 are larger than those of other cases. In comparison to Figure 12a and Figure 12b, the effect on GradT_{r} is stronger than on GradT_{a}.
Figure 13 shows the variations in GradT_{r} and GradT_{a} under Strategy II for different η. The results show that under different η, the trends of GradT_{r} and GradT_{a} with increasing HTCM do not change. Changing η has little effect on GradT_{r} and GradT_{a}.
Fig. 12 Variations in the radial and axial temperature gradients under Strategy I for different η. (a) GradT_{r} and (b) GradT_{a}. 
Fig. 13 Variations in the radial and axial temperature gradients under Strategy II for different η. (a) GradT_{r} and (b) GradT_{a}. 
3.3 Analysis of the asymmetrical segmented cooling characteristics
RPECT not only realizes the predefined roll profile but also switches among several roll profile states. This behaviour requires the ECR to be able to control the crown size and position. The symmetrical segmented cooling strategy can control crown size but cannot change the position of the maximum bulging point. Therefore, an asymmetric segmented cooling strategy is proposed. In this strategy, cooling areas I, II and III use different cooling intensities to adjust the maximum bulging amount. Table 2 shows examples of asymmetric segmented cooling strategies that are proposed to analyse the control ability. Cases A1∼A5 use a cooling strategy that changes the cooling intensity of one side cooling area. Cases A6∼A10 use a cooling strategy that changes the cooling intensities of one side cooling area as well as the middle cooling area.
Cooling intensity configuration scheme of the asymmetric segmented cooling strategy.
3.3.1 The effects of single side cooling and edge and middle cooling
Figure 14 shows the roll profile at 300 s under Cases A1∼A10. Changing HTCI can change the roll profile of the ECR and decrease the radial bulging of the corresponding area. For the transverse distribution of the bulging amount, the bulging amount of the ECR can be decreased with increasing HTCI, and the maximum bulging position gradually moves away from the centre of the ECR. When the cooling intensity is increased from 0.03 kW/(m^{2 }· K) to 3 kW/(m^{2 }· K), the maximum bulging value decreases by 1.7 µm, and the maximum bulging position shifts by 5 mm. The case where HTCI is 3 kW/(m^{2 }· K) has the most noticeable effect on crown movement. In Cases A6∼A10, the effect of the segmented cooling strategy is mainly to reduce the ECR bulging amount. Although it can cause asymmetric bulging on both sides of the ECR, directional movement of the maximum bulging amount is difficult to achieve. Therefore, to asymmetrically control the roll profile, changing the transverse position of the maximum bulging point necessitates the use of cooling strategies in Cases A1∼A5.
Fig. 14 The roll profile at 300 s under Cases A1∼A10. (a) Cases A1∼A5, and (b) Cases A6∼A10. 
3.3.2 The effect of single side cooling under different η
To analyse the effect of η on the asymmetric segmented cooling strategy, different values for η and different cooling parameters are introduced into the FE model, and the results are shown in Figure 15. The basic cooling intensity is 0.03 kW/(m^{2 }· K), and HTCI ranges from 0.03 kW/(m^{2 }· K) to 3 kW/(m^{2 }· K). The result shows that increasing HTCI causes the maximum bulging point to move away from the centre of the ECR and decreases the maximum bulging value.
Furthermore, increasing η can reduce the crown movement ability and the maximum bulging ability in the asymmetric segmented cooling strategy. When η is increased from 0.5 to 1, the maximum amount that the maximum bulging point shifts is 9.04 mm, 6.83 mm, and 3.98 mm, and the maximum bulging value decreases by 4.10 µm, 2.07 µm, and 0.55 µm. Therefore, in the asymmetric segmented cooling strategy, the maximum bulging point moves and maximum bulging value decreases simultaneously. By adjusting the value of η, the relationship between the maximum bulging position movement and the maximum bulging value can be coordinated.
To further analyse the roll profile control ability under different values for η and different basic cooling intensities, the maximum bulging point movement and the maximum bulging value are extracted, as shown in Figures 16 and 17. The results in Figure 16 show that by increasing the basic cooling intensity HTCB, the shifting amount of the maximum bulging point can be changed from positive to negative. When η is increased, the movement control range of the maximum bulging point can be reduced.
The results in Figure 17 show that increasing HTCI reduces the maximum bulging value of the ECR. Increasing the basic cooling intensity can also reduce the maximum bulging value. However, in Figure 17c , when η exceeds a certain value, the effects of HTCI on the maximum bulging value can be ignored.
Fig. 15 Variations in the maximum bulging amount and the position of the maximum bulging point with increasing HTCI. (a) The maximum shift of the maximum bulging point and (b) The maximum bulging amount. 
Fig. 16 The shifting amount of the maximum bulging point with increasing HTCI under different HTCB. (a) η = 0.5, (b) η = 0.67, and (c) η = 1. 
Fig. 17 The maximum bulging amount t with increasing HTCI under different HTCB. (a) η = 0.5, (b) η = 0.67, and (c) η = 1. 
3.4 Analysis of the sectional cooling adaptability of different ES structures
To further explore the roll profile control characteristics of different ES structures, three structure types and a 1000 mm long ECR are selected to analyse the structural adaptability of the segmented cooling strategies. The diameter of the ES is 100 mm, the length of the induction heating zone is 25 mm, the length of the middle contact zone is 50 mm, and the length of the edge contact zone is 25 mm. The ES structures are shown in Figure 18. Because the ECR roll length is much longer than the length of the ES, the end of the ECR exceeds the influence area of a single ES; thus, the bulging amount at the end of the roll is small, and the roll crown at this time can be considered the maximum bulging amount at the centre of the ECR. HTCB is set to 0.03 kW/(m^{2 }· K), and the variable cooling intensities are 0.03 kW/(m^{2 }· K), 1 kW/(m^{2 }· K) and 3 kW/(m^{2 }· K).
Figure 19 shows the variations in C2, GradT_{r} and GradT_{a} when HTCM is equal to HTCB and HTCE changes. Figure 19a shows that the ES structure has little influence on C2 as the cooling intensity changes, but it can change the overall bulging value. The maximum bulging value decreases as the length of the ES decreases. When structure I is used, C2 can be reduced, but the reduction is small. Figure 19b and c shows that structure II and structure III have no effect on the variation laws of GradT_{r} and GradT_{a}, but the longer the length of the ES, the larger the temperature gradient. When structure I is used, increasing HTCE leads to a significant increase in GradT_{r}, which also corresponds to a decrease in C2. Comprehensive analysis of the results reveals a considerable difference between the change law of structure I and that of structures II and III. The reason for this is that structure I has a shorter length than structures II and III, and the internal temperature field of the ECR is greatly affected by the edge cooling areas. When HTCE is increased, the middle area of the ECR in structure I can also be affected, resulting in a decline in the roll profile control ability.
Figure 20 shows the variations in C2, GradT_{r} and GradT_{a} when HTCE is equal to HTCB and HTCM changes. The results show that structural changes in the ES have little effect on C2, GradT_{r} and GradT_{a}. When the length of the ES increases, the control values of the parameters decrease, but they all follow the same variation law.
Fig. 18 Schematic diagrams of the ES structures. 
Fig. 19 Variations in the ECR status under Strategy I. (a) C2, (b) GradT_{r}, and (c) GradT_{a}. 
Fig. 20 Variations in the ECR status under Strategy II. (a) C2, (b) GradT_{r}, and (c) GradT_{a}. 
4 Conclusion
Symmetrically changing the cooling intensities of different cooling areas can adjust the roll profile of the ECR. The segmented cooling strategy that changes the cooling intensities of cooling areas I and III can increase the roll crown of the ECR. The segmented cooling strategy that changes the cooling intensity of cooling area II can decrease the roll crown of the ECR. The crown enhancement ability is weaker than the convexity weakening ability.
Asymmetrically changing the cooling intensities of different cooling areas can shift the position of the maximum bulging point. Two movement strategies are proposed in this paper. The movement strategy with a single cooling area has a greater impact than that of the movement strategy with the edge and middle cooling areas and can be used for roll crown shifting.
The component cooling ratio coefficient is proposed as an index to evaluate the segmented cooling strategy. Improving the coefficient impacts the different segmented cooling strategies, which needs to be considered during the segmented cooling process.
Segmented regulation characteristics exist in different multistage ES structures. Only when the length of the ES is small does the control law reverse, and the effect of this variation is relatively small.
Credit author statement
Tingsong Yang: Conceptualization, Methodology, Writing Reviewing and Editing. Yingwei Wang: Model analysis, Writing. Haijun Wang: Experiment, Methodology. Yang Hai: Experiment. Fengshan Du: Funding acquisition.
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This project is supported by the National Natural Science Foundation of China (Grant No. U1560206) and National Natural Science Foundation of China (Grant No. 51374184).
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Cite this article as: Tingsong Yang, Yingwei Wang, Haijun Wang, Yang Hai, Fengshan Du, Research on an external adjustment method for RPECT roll profiles based on the segmented cooling principle, Metall. Res. Technol. 119, 103 (2022)
All Tables
Cooling intensity configuration scheme of the asymmetric segmented cooling strategy.
All Figures
Fig. 1 Diagrammatic sketch of the division method of cooling areas of the surface of the ECR. 

In the text 
Fig. 2 RPECT electromagneticthermalstructural axisymmetric model with segmented cooling. 

In the text 
Fig. 3 Roll profile electromagnetic control experiment platform and the variation of the current. (a) roll profile electromagnetic control experiment platform and (b) the variation of the current. 

In the text 
Fig. 4 Temperature variation of ES induction area and the roll crown variation of ECR with time. (a) temperature variation, and (b) roll crown variation. 

In the text 
Fig. 5 Variations in C1, C2 and C under Strategy I. (a) C1, (b) C2 and (c) C. 

In the text 
Fig. 6 Variations of C1, C2 and C under Strategy II. (a) C1, (b) C2 and (c) C. 

In the text 
Fig. 7 Schematic diagram of the temperature gradient marking points. 

In the text 
Fig. 8 Variations of the radial and axial temperature gradients under Strategy I. (a) GradT_{r} and (b) GradT_{a}. 

In the text 
Fig. 9 Variations of the radial and axial temperature gradients under Strategy II. (a) GradT_{r} and (b) GradT_{a}. 

In the text 
Fig. 10 Variations in the values of C1, C2 and C under Strategy I for different η. (a) C1, (b) C2, and (c) C. 

In the text 
Fig. 11 Variations in the values of C1, C2 and C under Strategy II for different η. (a) C1, (b) C2, and (c) C. 

In the text 
Fig. 12 Variations in the radial and axial temperature gradients under Strategy I for different η. (a) GradT_{r} and (b) GradT_{a}. 

In the text 
Fig. 13 Variations in the radial and axial temperature gradients under Strategy II for different η. (a) GradT_{r} and (b) GradT_{a}. 

In the text 
Fig. 14 The roll profile at 300 s under Cases A1∼A10. (a) Cases A1∼A5, and (b) Cases A6∼A10. 

In the text 
Fig. 15 Variations in the maximum bulging amount and the position of the maximum bulging point with increasing HTCI. (a) The maximum shift of the maximum bulging point and (b) The maximum bulging amount. 

In the text 
Fig. 16 The shifting amount of the maximum bulging point with increasing HTCI under different HTCB. (a) η = 0.5, (b) η = 0.67, and (c) η = 1. 

In the text 
Fig. 17 The maximum bulging amount t with increasing HTCI under different HTCB. (a) η = 0.5, (b) η = 0.67, and (c) η = 1. 

In the text 
Fig. 18 Schematic diagrams of the ES structures. 

In the text 
Fig. 19 Variations in the ECR status under Strategy I. (a) C2, (b) GradT_{r}, and (c) GradT_{a}. 

In the text 
Fig. 20 Variations in the ECR status under Strategy II. (a) C2, (b) GradT_{r}, and (c) GradT_{a}. 

In the text 
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