4.1 The diameter of the roll and the diameter of the roll inner hole
To analyse the influence of the TEC layout on the bulging uniformity, the maximum bulging difference is defined as the CED, which is the difference between the maximum and minimum bulging amounts in the radial direction of the roll, and the calculation formula is shown in Formula (11).
where Cmax is the maximum bulging amount and Cmin is the minimum bulging amount.
Figure 7 shows the variation of Cmax and CED with DR under different Degs. The Tc is 50°C in these cases, and a uniform heat flow condition is established to provide a reference. In the uniform case, the element edges on the roll inner wall are set as the first thermal boundary with a value of 50 ℃. In Fig. 7 (a), Cmax can be increased with increasing DR and has similar change laws under different Degs. The law can be described as follows: with increasing DR, the increasing rates of Cmax gradually decrease. After the DR reaches 420 mm, Cmax begins to stabilize and no longer increases as the DR increases. Compared with the cases of different Degs, when Degs is decreased from 120° to 60°, Cmax can be gradually increased. Meanwhile, the influence value of DR on Cmax is 6.97 µm, 8.53 µm, 9.56 µm, and 10.3 µm, and the value is 10.88 µm in the uniform case. In Fig. 7 (b), CED can be gradually decreased as DR increases. When DR is large enough, CED is almost 0 µm. When the DR is small, the unevenness of roll bulging is more serious, and the larger the Degs is, the more obvious the bulging ratchet effect of ETCR. When Degs is reduced from 120° to 60°, the maximum CED values are 3.87 µm, 0.88 µm, 0.49 µm, and 0.09 µm, respectively. After Degs is reduced to 90°, the CED is less than 1 µm.
Figure 7 Variation in Cmax and CED with increasing DR under different Degs
In the ETCR, the reason for the roll profile variation is that the internal heat source changes the temperature field of the roll and further changes the thermal bulging effect. Therefore, the ratchet characteristic of the ETCR is related to the temperature field distribution. According to the result in Fig. 7, a severe bugling ratchet exists in the case with Degs = 120°. To select the evaluation criterion of the ratchet, the roll internal temperature fields with serious bulging ratchets are extracted, as shown in Fig. 8. Compared with the results in Fig. 8, the temperature range above 28°C can reflect the thermal ratchet of the ETCR. Considering that the corresponding cases in Fig. 8 are the worst cases of roll bulging uniformity, 28°C can be selected as the lowest temperature value of the thermal ratchet.
Figure 8 Roll internal temperature fields when Degs is 120°
Figure 9 shows the variation of the 28℃ temperature-affected zone inside the roll with DR under different Degs. The results show that with increasing Degs, the thermal ratchet is alleviated, which is consistent with the result in Fig. 7 (b). With a smaller DR, reducing Degs can alleviate the thermal ratchet problem caused by the TEC layout to a certain extent. When Degs is 60°, the thermal ratchet degree is lower, and uniform heat flow on the roll inner wall can be realized. With a larger roll diameter, except for the case where Degs is 120°, the ratcheting phenomenon of the temperature field in other cases can basically be eliminated.
Figure 9 Variation in the 28℃ temperature-affected zone under different DR
To further evaluate the thermal ratchet degree, the circumferential unit temperature rise control quantity is defined as ΔT. ΔT is the ratio of the temperature difference to Degs, and can be calculated by Formula (12).
where TDeg−max is the maximum temperature value within Degs, and TDeg−min is the minimum temperature value within Degs.
Figure 10 shows the change in ΔT of the inner wall and outer wall of the roll with changing DR under different Degs. In Fig. 10(a), ΔT can be decreased as DR increases, and the rate of decrease can be gradually decreased. Under the same DR, ΔT can be increased with increasing Degs. When Degs is changed from 60° to 120°, ΔT can be gradually increased with increasing DR, which indicates that increasing Degs can cause the ratchet phenomenon on the roll inner wall to become more serious. In Fig. 10 (b), the change in ΔT can be divided into two stages: the first stage is [140 mm, 220 mm], ΔT can be decreased rapidly with increasing DR, and the rate of decrease slowly declines. The second stage is [220 mm, 420 mm], the decrease rate declines further and finally stabilizes, and with the continuous increase in DR, ΔT finally approaches 0 ℃/°. Similarly, under the same DR, the increase in Degs can increase ΔT. When DR is 140 mm and Degs decreases from 120° to 60°, ΔT can decrease from 0.03 ℃/° to 0.01 ℃/°. The larger the Degs is, the larger the drop is. When Degs is 120°, the maximum drop is 0.03 ℃/°.
Figure 10 Variation in ΔT of the inner and outer walls of the roll with changing DR under different Degs
According to the above results, when Degs exceeds 90°, the ratio of β/α is too great, and the thermal ratchet phenomenon of the roll and the maximum difference value of roll bulging are both large, which is not suitable for ETCR. When Degs is less than 90°, the maximum difference value of roll bulging is small, and it can be further reduced by increasing DR, so it is suitable for ETCR.
In addition to the roll diameter, the diameter of the roll inner wall DIH is also a parameter that can affect the effectiveness of the ETCR. In Fig. 7, the cases where DR is 260 mm have good bulging ability and small CED and can be selected as the basic condition. To analyse the influence of DIH on ETCR, DIH and Degs are changed to analyse the bulging ability. Figure 11 shows the variation of Cmax and CED with DIH under different Degs. The results in Fig. 11 (a) show that with the increase of DIH, Cmax can be decreased, and the decrease rate of Cmax is the same under different Degs. Under the same DIH, the increase in Degs can reduce Cmax. When Degs is increased from 72° to 120°, the change in Cmax with increasing DIH is -1.13 µm, -1.37 µm, and − 1.4 µm. On the whole, under the same Tc, changing DIH has a small influence on Cmax, and the influence value is less than 2 µm. The results in Fig. 11 (b) show that CED gradually increases with increasing DIH. When Degs is 120°, the increased value of CED is the largest, and the value is 1.48 µm. When the Degs are 90° and 72°, the change values of CED are 0.29 µm and 0.11 µm, respectively.
Figure 11 Variation in Cmax and CED with increasing DIH under different Degs
In addition, the results in Fig. 11 (a) also show that the change trend of Cmax in the uniform case is different from those in other cases. Cmax can be gradually increased with increasing DIH in the uniform case, while Cmax can be decreased with increasing DIH in the other cases. The case with Degs of 120° is the most serious. To analyse the reason, the node temperature, which is located in the radial path from the roll inner wall to the maximum bulging point of the roll surface, is extracted when Degs is 120°, as shown in Fig. 12. The results show that the larger the DIH is, the lower the point temperature at the same distance from the roll inner wall, so the corresponding thermal bulge is lower at the same position. The reason is that under a constant roll diameter, the expansion of DIH is equivalent to reducing the roll wall thickness. The area close to the roll surface can obtain a higher temperature and form a larger thermal bulge. However, this law exists when there is no difference or a small difference in the roll circumferential temperature. When the heat source is uniformly distributed in the circumferential direction, the bulging ability can be improved by decreasing the roll wall thickness. In addition to the uniform case, the heat source is nonuniform in the circumferential direction, so increasing DIH reduces the roll wall thickness and increases the distances among TECs. For two adjacent TECs, the symmetry plane between two TECs is also the symmetry plane of the heat transfer influence zone. With increasing DIH, the circumferential heat transfer between the adjacent TEC is also more obvious, which leads to a decrease in Cmax.
Figure 13 shows the internal temperature fields of the roll with changing DIH. The results show that in Fig. 13(a), (d), and (g), when Degs is 120°, regardless of the DIH value, there is a severe thermal ratchet problem in the internal temperature field of the roll. When Degs is reduced to 100°, the temperature ratchet is relieved, but when DIH is larger, such as Fig. 13 (h), there is still a severe thermal ratchet. If Degs is further reduced, the thermal ratchet in the case in which the DIH is 120 mm can also be relieved. The above changes are because the reduction in Degs can increase the total direct influence angle, so the input heat flow distribution on the roll inner wall is more uniform. In addition, it is not easy to produce thermal ratchet. When DIH is increased, the total direct influence angle can be decreased and the indirect influence angle can be increased, so the uniformity of the input heat flow can be decreased on the roll inner wall, and the thermal ratchet effect is aggravated.
Figure 13 Variation of the roll temperature field under different DIH and Degs
Figure 14 shows the change in ΔT of the inner and outer walls with DIH under different Degs. In Fig. 14 (a), the ΔT of the inner wall of the roll can be increased with increasing DIH, but the growth rate gradually decreases. Under the same DIH, the larger the Degs, the larger the ΔT, and the worse the uniformity of the heat flow in the roll inner wall. In Fig. 14 (b), the ΔT of the roll outer wall can also be increased with increasing DIH, but the growth rate gradually increases. The difference in the growth rate of ΔT between Fig. 14 (a) and (b) is that increasing DIH can reduce the distance between the heat source and the roll surface, and the thermal ratchet caused by different Degs is more likely to appear on the outer wall of the roll.
Figure 14 Variation in ΔT of the inner and outer walls of the roll with changing DIH under different Degs
In summary, increasing DR can increase Cmax and decrease CED, while increasing DIH can decrease Cmax and increase CED. Changing Degs can affect the effects of DR and DIH. Comparing the results of Fig. 7(b) and Fig. 11(b), the cases in which Degs is less than 90° have a relatively small maximum difference value of roll bulging, so these cases are more suitable for ETCR than the cases in which Degs is more than 90°. Because Degs is the parameter of TEC, it is necessary to further analyse the influence of TEC parameters on the effect of ETCR.
4.2 The TEC control temperature
According to previous research results, the bulging control effect of the roll is more obvious and CED is small when DR is 260 mm and DIH is 100 mm. Therefore, these parameters are selected to analyse the influence of Tc on the thermal ratchet effect. Figure 15 shows the variation of Cmax and CED with Tc under different Degs. The results show that both Cmax and CED can increase linearly with increasing Tc. In Fig. 15(a), when Degs is reduced from 120° to 60°, the growth rate of Cmax is 0.25 µm/℃, 0.32 µm/℃, 0.38 µm/℃, and 0.43 µm/℃. Compared with the uniform case of 0.48 µm/℃, the smaller the value of Degs is, the closer the bulging effect of the roll is to the uniform case. In Fig. 15(b), in addition to the case in which Degs is 120°, other cases have lower CED. When Degs is decreased from 90° to 60°, the variation rate of CED is 0.005 µm/℃, 0.001 µm/℃, and 0.0001 µm/℃. The results indicate that cases in which Degs are 90°, 72°, and 60° can meet the requirements of uniform circumferential bulging.
Figure 15 Variation in Cmax and CED with increasing Tc under different Degs
Figure 16 shows the changes in the 28℃ temperature-affected zone with Tc under different Degs. The results show that when Degs is 120°, only the case with a larger Tc has a relatively small circumferential difference in the temperature field. In comparison, a case with a lower Tc has a more severe thermal ratchet phenomenon. Compared to the results in Fig. 15(b), even if Tc is increased, the cases with Degs of 120° are still not suitable as an optional form of TEC layout. When Degs is 90°, 72°, and 60°, the thermal ratchet phenomenon occurs only when the TEC control temperature is small, and the thermal ratchet phenomenon is eliminated after slightly increasing Tc.
Figure 16 Variation of the 28℃ temperature-affected zone with Tc under different Degs
Figure 17 shows the change in ΔT of the inner and outer walls of the roll with Tc under different Degs. The results show that the ΔT of the outer wall and inner wall also increases linearly with increasing Tc. In Fig. 17 (a), the growth rate of ΔT increases with increasing Degs, but the difference is not large. In the cases, the growth curves of Degs 90° and 120° almost coincide in value and trend, indicating that the heat flow uniformities of the roll inner wall under the two cases are approximately the same. On the other hand, under the same Tc, ΔT can be increased with increasing Degs, and the heat flow uniformity of the roll inner wall is worse. In Fig. 17 (b), the change value of ΔT in the roll outer wall with changing Tc is much smaller than that of the roll inner wall. The change rule of ΔT with Tc and Degs is the same as in Fig. 17(a). Therefore, in the cases of different Tc, the temperature uniformity of the roll inner wall is greatly affected by Tc, while the influence on the roll outer wall is very small. For the roll inner wall temperature, the increase in Tc and Degs can lead to an increase in the circumferential unevenness of the roll inner wall temperature.
Figure 17 Variation in ΔT of the inner and outer walls of the roll with changing Tc under different Degs
4.3 The single piece influence angle
In addition to Tc, the TEC amount is also an important parameter of TEC. The electronic temperature-controlled roll's circumferential sheet-carrying capacity is related to the diameter of the inner hole of the roll. Therefore, the parameters of the FE model are as follows: DR is selected as 260 mm, DIH is 80 mm, 100 mm, and 120 mm, and Tc is selected as 50°C. Figure 18 shows the variation in Cmax and CED with Degs under different DIH. The results showed that under different DIH, Cmax can be decreased and CED can be increased with increasing Degs. The change in CED is divided into two stages: when Degs is [40°, 90°], the difference in CED under different DIH is small. It can be considered that changing DIH cannot affect the effect of Degs on the radial bulging unevenness. When Degs exceeds 90°, the greater the DIH is, the more serious the radial bulging unevenness of the roll.
Figure 18 Variation of Cmax and CED with Degs under different DIH
Figure 19 shows the variation in the roll internal temperature field under different DIH and Degs. The results show that, regardless of DIH, with the increase in Degs, the thermal ratchet phenomenon can become increasingly obvious. Especially when Degs is 120°, the thermal ratchet phenomenon is so obvious that it affects the roll bulging ability and the circumferential uniformity of the roll bulging. Meanwhile, this effect can further increase as DIH increases. It can be seen from the temperature field distribution that the smaller the DIH is, the greater the Degs demand value that can ensure uniform bulging is. For example, for a case in which DR is 80 mm, the Degs demand value is 90°; for a case in which DR is 100 mm, the Degs demand value is 72°; and for a case in which DR is 120 mm, the Degs demand value is 60°.
Figure 19 Variation of the roll temperature field under different DIH and Degs
Figure 20 shows the change in ΔT of the inner and outer walls of the roll with changing Degs under different DIH. In Fig. 20(a), the changing trend of ΔT can be described as "increasing first and then becoming stable." The smaller the DIH is, the smaller the ΔT on the inner wall of the roll, and the better the uniformity of the inner wall heat flow. In Fig. 20 (b), ΔT increases with Degs, but its value is much smaller than that in Fig. 20(a); that is, the ΔT in the roll outer wall is smaller, and the temperature distribution is more uniform.
Figure 20 Variation in ΔT of the inner and outer walls of the roll with changing Degs under different DIHs