## 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 *C**ED*, 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).

$${C}_{ED}={C}_{max}-{C}_{min}$$

11

where *C**max* is the maximum bulging amount and *C**min* is the minimum bulging amount.

Figure 7 shows the variation of *C**max* and *C**ED* with *D**R* under different *Deg**s*. The *T**c* 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), *C**max* can be increased with increasing *D**R* and has similar change laws under different *Deg**s*. The law can be described as follows: with increasing *D**R*, the increasing rates of *C**max* gradually decrease. After the *D**R* reaches 420 mm, *C**max* begins to stabilize and no longer increases as the *D**R* increases. Compared with the cases of different *Deg**s*, when *Deg**s* is decreased from 120° to 60°, *C**max* can be gradually increased. Meanwhile, the influence value of *D**R* on *C**max* 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), *C**ED* can be gradually decreased as *D**R* increases. When *D**R* is large enough, *C**ED* is almost 0 µm. When the *D**R* is small, the unevenness of roll bulging is more serious, and the larger the *Deg**s* is, the more obvious the bulging ratchet effect of ETCR. When *Deg**s* is reduced from 120° to 60°, the maximum *C**ED* values are 3.87 µm, 0.88 µm, 0.49 µm, and 0.09 µm, respectively. After *Deg**s* is reduced to 90°, the *C**ED* 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 *Deg**s* = 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 *D**R* under different *Deg**s*. The results show that with increasing *Deg**s*, the thermal ratchet is alleviated, which is consistent with the result in Fig. 7 (b). With a smaller *D**R*, reducing *Deg**s* can alleviate the thermal ratchet problem caused by the TEC layout to a certain extent. When *Deg**s* 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 *Deg**s* 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** D*R*

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 *Deg**s*, and can be calculated by Formula (12).

$${\varDelta T}_{c}=\frac{{T}_{Deg-max}-{T}_{Deg-min}}{{Deg}_{s}}$$

12

where *T**Deg−max* is the maximum temperature value within *Deg**s*, and *T**Deg−min* is the minimum temperature value within *Deg**s*.

Figure 10 shows the change in *ΔT* of the inner wall and outer wall of the roll with changing *D**R* under different *Deg**s*. In Fig. 10(a), *ΔT* can be decreased as *D**R* increases, and the rate of decrease can be gradually decreased. Under the same *D**R*, *ΔT* can be increased with increasing *Deg**s*. When *Degs* is changed from 60° to 120°, *ΔT* can be gradually increased with increasing *D**R*, which indicates that increasing *Deg**s* 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 *D**R*, 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 *D**R*, *ΔT* finally approaches 0 ℃/°. Similarly, under the same *D**R*, the increase in *Deg**s* can increase *ΔT*. When *D**R* is 140 mm and *Deg**s* decreases from 120° to 60°, *ΔT* can decrease from 0.03 ℃/° to 0.01 ℃/°. The larger the *Deg**s* is, the larger the drop is. When *Deg**s* 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 *Deg**s* 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 *Deg**s* is less than 90°, the maximum difference value of roll bulging is small, and it can be further reduced by increasing *D**R*, so it is suitable for ETCR.

In addition to the roll diameter, the diameter of the roll inner wall *D**IH* is also a parameter that can affect the effectiveness of the ETCR. In Fig. 7, the cases where *D**R* is 260 mm have good bulging ability and small *C**ED* and can be selected as the basic condition. To analyse the influence of *D**IH* on ETCR, *D**IH* and *Deg**s* are changed to analyse the bulging ability. Figure 11 shows the variation of *C**max* and *C**ED* with *D**IH* under different *Deg**s*. The results in Fig. 11 (a) show that with the increase of *D**IH*, *C**max* can be decreased, and the decrease rate of *C**max* is the same under different *Deg**s*. Under the same *D**IH*, the increase in *Deg**s* can reduce *C**max*. When *Deg**s* is increased from 72° to 120°, the change in Cmax with increasing *D**IH* is -1.13 µm, -1.37 µm, and − 1.4 µm. On the whole, under the same Tc, changing *D**IH* has a small influence on Cmax, and the influence value is less than 2 µm. The results in Fig. 11 (b) show that *C**ED* gradually increases with increasing *D**IH*. When *Deg**s* is 120°, the increased value of *C**ED* is the largest, and the value is 1.48 µm. When the *Deg**s* are 90° and 72°, the change values of *C**ED* 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 *C**max* in the uniform case is different from those in other cases. *C**max* can be gradually increased with increasing *D**IH* in the uniform case, while *C**max* can be decreased with increasing *D**IH* in the other cases. The case with *Deg**s* 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 *Deg**s* is 120°, as shown in Fig. 12. The results show that the larger the *D**IH* 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 *D**IH* 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 *D**IH* 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 *D**IH*, the circumferential heat transfer between the adjacent TEC is also more obvious, which leads to a decrease in *C**max*.

Figure 13 shows the internal temperature fields of the roll with changing *D**IH*. The results show that in Fig. 13(a), (d), and (g), when *Deg**s* is 120°, regardless of the *D**IH* value, there is a severe thermal ratchet problem in the internal temperature field of the roll. When *Deg**s* is reduced to 100°, the temperature ratchet is relieved, but when *D**IH* is larger, such as Fig. 13 (h), there is still a severe thermal ratchet. If *Deg**s* is further reduced, the thermal ratchet in the case in which the *D**IH* is 120 mm can also be relieved. The above changes are because the reduction in *Deg**s* 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 *D**IH* 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 *D**IH* under different *Deg**s*. In Fig. 14 (a), the *ΔT* of the inner wall of the roll can be increased with increasing *D**IH*, but the growth rate gradually decreases. Under the same *D**IH*, the larger the *Deg**s*, 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 *D**IH*, but the growth rate gradually increases. The difference in the growth rate of *ΔT* between Fig. 14 (a) and (b) is that increasing *D**IH* can reduce the distance between the heat source and the roll surface, and the thermal ratchet caused by different *Deg**s* 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 *D**R* can increase *C**max* and decrease *C**ED*, while increasing *D**IH* can decrease *C**max* and increase *C**ED*. Changing *Deg**s* can affect the effects of *D**R* and *D**IH*. Comparing the results of Fig. 7(b) and Fig. 11(b), the cases in which *Deg**s* 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 *Deg**s* is more than 90°. Because *Deg**s* 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 *C**ED* is small when *D**R* is 260 mm and *D**IH* is 100 mm. Therefore, these parameters are selected to analyse the influence of *T**c* on the thermal ratchet effect. Figure 15 shows the variation of Cmax and CED with *T**c* under different *Deg**s*. The results show that both *C**max* and *C**ED* can increase linearly with increasing *T**c*. In Fig. 15(a), when *Deg**s* is reduced from 120° to 60°, the growth rate of *C**max* 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 *Deg**s* 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 *Deg**s* is 120°, other cases have lower *C**ED*. When *Deg**s* is decreased from 90° to 60°, the variation rate of *C**ED* is 0.005 µm/℃, 0.001 µm/℃, and 0.0001 µm/℃. The results indicate that cases in which *Deg**s* 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 *T**c* under different *Deg**s*. The results show that when *Deg**s* is 120°, only the case with a larger *T**c* has a relatively small circumferential difference in the temperature field. In comparison, a case with a lower *T**c* has a more severe thermal ratchet phenomenon. Compared to the results in Fig. 15(b), even if *T**c* is increased, the cases with *Deg**s* of 120° are still not suitable as an optional form of TEC layout. When *Deg**s* 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 *T**c*.

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 *T**c* under different *Deg**s*. The results show that the *ΔT* of the outer wall and inner wall also increases linearly with increasing *T**c*. In Fig. 17 (a), the growth rate of *ΔT* increases with increasing *Deg**s*, but the difference is not large. In the cases, the growth curves of *Deg**s* 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 *T**c*, *ΔT* can be increased with increasing *Deg**s*, 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 *T**c* is much smaller than that of the roll inner wall. The change rule of *ΔT* with *T**c* and *Deg**s* is the same as in Fig. 17(a). Therefore, in the cases of different *T**c*, the temperature uniformity of the roll inner wall is greatly affected by *T**c*, while the influence on the roll outer wall is very small. For the roll inner wall temperature, the increase in *T**c* and *Deg**s* 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 *T**c*, 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: *D**R* is selected as 260 mm, *D**IH* is 80 mm, 100 mm, and 120 mm, and *T**c* is selected as 50°C. Figure 18 shows the variation in *C**max* and *C**ED* with *Deg**s* under different *D**IH*. The results showed that under different *D**IH*, *C**max* can be decreased and *C**ED* can be increased with increasing *Deg**s*. The change in *C**ED* is divided into two stages: when *Deg**s* is [40°, 90°], the difference in *C**ED* under different *D**IH* is small. It can be considered that changing *D**IH* cannot affect the effect of *Deg**s* on the radial bulging unevenness. When *Deg**s* exceeds 90°, the greater the *D**IH* 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 *D**IH* and *Deg**s*. The results show that, regardless of *D**IH*, with the increase in *Deg**s*, the thermal ratchet phenomenon can become increasingly obvious. Especially when *Deg**s* 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 *D**IH* increases. It can be seen from the temperature field distribution that the smaller the *D**IH* is, the greater the *Deg**s* demand value that can ensure uniform bulging is. For example, for a case in which *D**R* is 80 mm, the *Deg**s* demand value is 90°; for a case in which *D**R* is 100 mm, the *Deg**s* demand value is 72°; and for a case in which *D**R* is 120 mm, the *Deg**s* 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 *Deg**s* under different *D**IH*. In Fig. 20(a), the changing trend of *ΔT* can be described as "increasing first and then becoming stable." The smaller the *D**IH* 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 *Deg**s*, 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