Research on Temperature Change Law and Non-Uniform Distribution Characteristics of Electromagnetic Control Roll Based on Rotating Heat Flow

Abstract

The uniform temperature distribution on the surface of the electromagnetic control roll (ECR) has a great impact on the quality of the strip; therefore, temperature control is essential. In order to study this issue, a two-dimensional volume of fluid (VOF) model was established using the simulation software FLUENT (2024 R1) to analyze the radial cooling capacity and surface temperature uniformity of the ECR under different process parameters, and an experimental validation was carried out at the same time. The error between the experiment and the model was less than 5% of the maximum temperature, proving the model is accurate. The results of the analysis show that the use of a controlled temperature mode has an effect on the cooling capacity and the speed has no effect on the cooling capacity. The temperature difference between the two sides of the ECR is too large, which will make the uniformity of the ECR surface temperature worse. While too high or too low, a roll speed and coolant injection speed will increase the non-uniformity of the ECR surface temperature; when the roll speed is 12 rad/s or coolant injection speed is 5 m/s, the ECR surface temperature distribution uniformity is the best. Properly adjusted process parameters can improve the cooling performance and ECR surface temperature uniformity.

1. Introduction

Metal strips are an indispensable basic raw material for manufacturing, and their quality and capacity demands continue to rise with the technological development of the manufacturing industry. These demands have gradually evolved into the demand orientations of wide, thin, and high-value [1,2,3]. In contrast to the relatively straightforward production of low-width end strips, high-width end strips are prone to high flatness and thickness fluctuations in the production process. Flatness control technology is commonly used in the field of strip manufacturing in order to solve the high number of flatness problems that occur in production and to improve the production capacity of high-end strips [4]. The core of flatness control is the reasonable regulation of the roll gap shape to improve the product quality. At present, roll gap rigidity control is widely used in the production line, which usually includes the use of external devices to adjust the roll gap of the flatness control technology, such as roll bending [5] and roll shifting [6], according to the particular roll profile and external device, such as CVC technology [7], as well as the use of flatness control technologies that adjust the roll gap using an internal driving device, such as edge drop control [8]. However, the roll gap rigidity adjustment technology often suffers from low adjustment accuracy and inflexible adjustments in practical applications [9], which greatly restricts high-precision strip-rolling. The use of roll profile electromagnetic control technology (RPECT) as the head of the roll gap flexible control technology can allow for the flexible, precise adjustment of the roll axial sections that affect the roll profile’s topography to achieve micro-scale adjustments [10]. The technology allows for the flexible control of the roll gap through the thermal force expansion of the RPECT element, in which the electromagnetic stick (ES) is the core component of the thermal expansion. The size design and temperature control achieved in this way can allow for a more accurate control of the overall roll profile [11]. There has been a great deal of research on RPECT, and scholars have proposed basic concepts and modelled experimental validations [12]. Additionally, various aspects of the feasibility and controllability of RPECT have been explored by scholars, such as explorations of the influence of thermal force roll profiles through changes in the structural dimensions, their structural arrangement, and their process parameters (carry current, cooling temperature) [13,14,15]. In fact, the electromagnetic control roll (ECR) in the rotary heat transfer mechanism is affected by both the external cooling mechanism and the internal heating mechanism. This means that the uniformity of the ECR surface temperature is also critical to the thermal contribution of the roll profile. The differences in the shape of each zone of the heat-contributing roll profile result from an uneven ECR surface temperature, which will weaken the control of the overall roll profile, and ultimately lead to inconsistency between the theoretically derived results and the actual results. Therefore, how to strengthen the force contribution profile and weaken the thermal contribution profile are the focus in this paper.

The core control element of ECR is the ES; in the process of ECR rotary heat transfer, internal ESs will produce a constant heat source from the inside out for heat conduction. During the ECR’s internal heat conduction, the ECR surface will be subjected to a coolant spray to form a convection heat transfer and ensure that the internal heat is constantly reduced. The mixture of heat convection and heat conduction affects the cooling capacity of the ECR and the uniformity of the temperature distribution across the ECR surface. Related scholars have conducted research on the effect of different control strategies on temperature uniformity. Wei et al. [16] investigated the temperature distribution uniformity within the cathode catalytic layer (CCL) in a proton exchange membrane fuel cell (PEMFC) and its effect on the power output and found that the Pt gradient distribution at different voltages has an opposite effect on the power output as well as the temperature distribution uniformity. Yang et al. [17] used CFD numerical simulation and experiments to investigate the effect of an oven’s structural parameters on the uniformity of the temperature field inside the oven. It was found that modifying the structure of the oven tailgate, adjusting the distribution of the air volume, and modifying the distribution of the air outlets can significantly enhance the uniformity of the temperature field inside the oven. Long et al. [18] enhanced the heat transfer inside a tube and improved the uniformity of the temperature distribution by incorporating novel deflector vanes inside a solar external receiver tube. Zhu et al. [19] fabricated a new type of heat pipe with a grooved porous structure and conducted an analysis of the temperature uniformity and temperature response with varying core widths and powder sizes. Shi et al. [20] used a multi-physics coupled finite element method to simulate the dynamic induction heating process, and conducted an analysis of the effects of three parameters, namely the moving speed of the induction coil, the gap between the induction coil and the workpiece, and the width of the induction coil, on the uniformity of the temperature distribution in the cam thickness direction and the contour line direction. Tu et al. [21] demonstrated, both theoretically and experimentally, that improvements in the uniformity of the temperature distribution can be achieved by increasing the porosity and decreasing the diameter of the fluid channels. Luo et al. [22] established a two-dimensional, coupled, heat engine, finite element model and proposed two evaluation indexes, the standard deviation of equivalent plastic strain and the standard deviation of temperature, with the objective of addressing the issue of the frequent macro- and micro-defects resulting from the significant inhomogeneity of strain and temperature distributions during the forging process of Ti-6Al-4V turbine blades. It was determined that a reasonable combination of process parameters, which considers flow resistance, flow localization, deformation, and frictional heating effects, is crucial for achieving temperature uniformity during the forging of titanium alloy blades. Song et al. [23] conducted a comprehensive analysis of the eddy current and temperature fields of the TFIH device with both the original and the new magnetic poles under identical excitation conditions. This was achieved through the utilization of the magneto–thermal coupling calculation method. The findings revealed that the newly designed magnetic poles have the potential to markedly enhance the uniformity of temperature distribution across the surface of the strip while simultaneously reducing magnetoresistance within the magnetic circuit. While the studies above have not examined the temperature distribution and cooling performance of the ECR surface, they demonstrate that modifying the device structure or process parameters can enhance the heat transfer characteristics of the device, thereby achieving a more uniform temperature distribution.

Accordingly, in the present study, the magneto-thermal driving principle of ECR was analyzed to develop a two-dimensional VOF fluid simulation model for the cooling performance of ECR under the dual action mechanism of external cooling and internal heat conduction, as well as the distribution of surface temperature uniformity. In this study, the cooling effect and surface temperature uniformity of ECR were examined, discussing the influence of various process parameters, including coolant temperature (T₁), internal heat source temperature (T₂), initial roll temperature (T₃), coolant injection speed (v), and roll speed (w).

2. Methods

2.1. Electromagnetic Control Roll (ECR) Magneto-Thermal Drive Principle

Figure 1 shows the specific structure of the ECR and the roll profile control principle of the RPECT. The RPECT implementation framework is comprised primarily of the following components: the electromagnetic stick (ES), the induction coil (IC), the polymagnet (PL), and the ECR. In this configuration, each group of ESs can be driven in the IC to generate a local heat source, which gradually forms a local temperature field. This field is then transported through the ES contact area from the inside to the outside of the heat transfer. The inductively heated zone serves as the internal heat source for the ECR, and the impact of roll profile control can be influenced by modifying the induction zone’s temperature level and heat transfer capability. The multiple ES and ECR temperature differences lead to the formation of two distinct mechanisms, the ES thermal expansion mechanism and the ECR internal constraint mechanism. These two mechanisms act on the roll bar contact surface, generating contact pressure and driving the roll force profile. As a consequence of the constraint mechanism in the roll, the thermal profile of the roll is also formed within the roll itself. The local temperature field initiates this within the roll and constitutes a thermal hybrid drive in conjunction with the roll force profile.

In the ECR heat transfer process, the electromagnetic and temperature fields are mainly involved. The internal ES temperature, convective heat transfer from air and coolant to the ECR surface, radiative heat transfer between the environment and the roll, and heat conduction within the roll influence the temperature field. Firstly, the internal induced heat source is described by a system of Maxwell equations, in which there is a bidirectional coupling between the electromagnetic field and the temperature field. The specific calculation formula is shown in Equation (1) [24,25], as follows:

$$\begin{array}{l}\nabla \times \overrightarrow{H}=\overrightarrow{J}+{\displaystyle \frac{\partial \overrightarrow{D}}{\partial t}}\\ \nabla \times \overrightarrow{E}=-{\displaystyle \frac{\partial \overrightarrow{E}}{\partial t}}\\ \nabla \cdot \overrightarrow{B}=0\\ \nabla \cdot \overrightarrow{D}=\rho \end{array}$$

(1)

where $\overrightarrow{H}$ and $\overrightarrow{E}$ denote the magnetic and electric field strength; $\overrightarrow{B}$ and $\overrightarrow{D}$ denote the flux density and potential shift vector; $\overrightarrow{J}$ is the current density; and $\rho$ is the bulk charge density. Based on these equations, the distribution of the electromagnetic field can be obtained, and the heat source generated by induction heating can be calculated, as in the following Equation (2):

$$Q=\overrightarrow{J}\cdot \overrightarrow{E}$$

(2)

where Q denotes the induced heat source in J/m³. The induced heat source Q causes a change in the temperature field, which can be converted to a temperature T by the conservation of energy and the heat conduction equation, as follows in Equation (3):

$$\psi c_p{\displaystyle \frac{\partial T}{\partial t}}=k{\nabla }^{2}T+\overrightarrow{J}\cdot \overrightarrow{E}$$

(3)

where $\psi$ denotes the density of the material; $c_p$ denotes its specific heat capacity of the material; k denotes the thermal conductivity of the material; and T denotes the temperature.

The ECR is used as the internal heat source of the roll through the inductive heat source generated after the internal ES is energized. Different current strengths can be used to obtain different internal heat source temperatures in the experiments. To simplify the model, in this study, a mechanism analysis is used to replace the process effect of the current strength with the temperature of the internal heat source. This equivalence has provided a more intuitive characterization of the impact of the internal heat source on the temperature distribution within the ECR.

Regarding the temperature field under the external cooling mechanism, the ECR is subjected to a convective heat transfer between the air and coolant to the roll, consistent with Newton’s law of cooling, as follows in Equation (4):

$$\begin{array}{l}{q}_{\mathrm{conv}1}=h_1(T_s-T_f)\\ q_{\mathrm{conv}2}=h_2(T_s-T_{\infty })\end{array}$$

(4)

where $q_{\mathrm{conv}1}$ is the convective heat transfer with the coolant; $q_{\mathrm{conv}2}$ is the convective heat transfer with the air; $h_1$ is the convective heat transfer coefficient with the coolant; $h_2$ is the convective heat transfer coefficient with the air, and the variation in the $h_2$ value is negligible in the range of working conditions in this study. As the reference indicates, the convective heat transfer coefficient between the ECR and the surrounding environment is typically 12 to 59 W/(m²·K). The radiation heat transfer between the environment and the roll is negligible because it is tiny compared to the convective heat transfer.

The internal temperature field of the ECR is influenced by the thermal conduction of the internal electromagnetic heat source and satisfies Fourier’s law of thermal conductivity, as follows in Equation (5):

$$q_{\mathrm{cond}}=-k\Delta T$$

(5)

where $q_{\mathrm{cond}}$ is the amount of heat conduction; k is the material thermal conductivity; and $\Delta T$ is the temperature difference.

The above analysis shows that the thermal force drive model can significantly improve the flatness and thickness problems in metal strip production while bringing some new issues. The roll’s thermal expansion profile is subject to real-time temperature fluctuations, resulting in a dynamic range of variation. The thermal contribution profile is considerable, leading to frequent flatness problems. The current limitations of RPECT technology in flexibly and accurately regulating the thermal contribution profile have prompted the investigation of potential solutions to mitigate the adverse effects of this profile on the actual production process while maintaining the uniformity of the ECR surface temperature distribution.

2.2. Model Building

The roll heat-flow-coupled cooling system mainly includes the ECR, internal heat source ES, and cooling nozzle. In the Fluent simulation model, the turbulence is modeled using the RNG k-ε model, the energy equation is activated with the gravity term, the wall is represented using the standard wall function model, the air is designated as the primary phase, and the coolant is designated as the secondary phase to construct the two-dimensional heat-flow coupled VOF model. The algorithm uses a semi-implicit SIMPLE method to resolve the system of pressure-coupled equations, which is used to simulate the injection of coolant into the roll face of the roll. Due to the high computational accuracy required for multiphase flow problems, the residuals of the velocity equation, the turbulence equation, and the VOF term are all guaranteed to be 1 × 10⁻⁵. Due to the phenomenon of forced convection heat transfer between the coolant and the ECR surface, the grid of the coolant in contact with the ECR surface is encrypted, and the entire spray zone grid is refined, as shown in Figure 2. In order to gain further insight into the impact of varying process parameters, the cooling control mode proposed in this paper involves a continuous spray with a control time of 300 s.

In the simulation model, the diameter of the ECR is $\varphi$ 400 mm, the diameter of the ECR bore is $\varphi$ 75 mm, and the material of both the ECR and ES is C45 steel (i.e., 1045 steel in the SAE). The coolant inlet nozzles are oriented at 45° angles to the right and left of the horizontal reference line. The thermophysical parameters of the material, including thermal conductivity and specific heat capacity, exhibit a continuous change with temperature. The thermophysical parameter curves are illustrated in Figure 3. The current density value range is 3–5 A/mm², with a fundamental value of 4 A/mm². The current frequency range is 400–3000 Hz, with a fundamental value of 400 Hz. The temperature of the internal heat source is verified to be 373 K.

2.3. Model Verification

In order to ascertain the veracity of the 2D VOF model, an experimental platform for the roll cooling system was constructed, as shown in Figure 4. The experimental platform comprises the following three principal components: an internal heat source power control module, a cooling control module, and a temperature detection module. For the internal heat source power control module, the inverter power supply can provide current input to the IC and collect the temperature signal from the internal heat source to control the internal temperature. The cooling control module can control different cooling process parameters to regulate the heat transfer capability between the ECR and the coolant, as well as the environment. Meanwhile, the temperature detection module is used to collect the temperature of the circumferential sampling points of the roll by placing thermocouples at the circumferential surface located in the central cross-section of the ECR and then verifying whether the simulation results are correct. The structural parameters of the ECR are as follows: the radius of the ECR is 200 mm, and the radius of the inner hole of the ECR is 75 mm. The current density and the frequency preset values are 4 A/mm² and 400 Hz, respectively. These values have been determined with the support of the aforementioned functional modules. The control coolant temperature T₁ is 298 K; the initial temperature T₂ of the ECR is 303 K; the heat source temperature T₃ of the ECR thermal mixing drive is 373 K and the coolant injection speed v is 5 m/s; the roll speed w is 0 rad/s; and the single-valve nozzle is opened for experimental verification.

A comparison of the experimental and simulation results of surface temperature in the central cross-section of the ECR is shown in Figure 5. The results showed that the simulation and experimental results compared to the first 60s difference are slightly larger, and the 60–120 s ECR surface temperature of the two are the same; the two different conditions of the ECR surface temperature error rate are a maximum of 5% or so. This indicates that the accuracy and precision of the model are sufficient to meet the demands of this study and can be used to investigate the temperature uniformity of the ECR under different process parameters. The impact of the disparate working conditions on temperature was also corroborated in this study; however, the circumstances remain largely analogous and thus will not be reiterated.

2.4. Guidelines for the Evaluation of Model

In order to more accurately assess the impact of the varying process parameters on the radial temperature distribution of the roll and the uniformity of the circumferential surface temperature distribution in the central cross-section of the ECR, in this study, three concepts are proposed of the ECR surface temperature extremes (ε), the circumferential average temperature gradient (γ), and the radial temperature cooling depth layer (h). The radial temperature cooling depth layer means, for a moment, the roll by the outside world after cooling, the roll internal temperature reduction radial depth, the cooling depth layer is larger, indicating that the external cooling mechanism on the internal cooling influence area is more significant, that is, the cooling effect is more effective. The three are calculated as follows, in Equations (6)–(8):

$$\epsilon =T_{\max }-T_{\min }$$

(6)

$$\gamma ={\displaystyle \frac{{\displaystyle \sum _{i=1}^N\frac{T_i}{ra{d}_i}}}{N}}$$

(7)

$$h=X_{TS}-X_{T0}$$

(8)

where $T_{\max }$ is the maximum value of ECR surface temperature; $T_{\min }$ is the minimum value of ECR surface temperature; $T$ is the temperature at a point on the ECR surface; $X_{TS}$ is the radial distance corresponding to the ECR surface temperature; $X_{T0}$ is the maximum radial distance corresponding to the initial temperature of the roll; and $ra{d}_i$ is the angle value corresponding to the temperature at a point on the ECR surface.

3. Numerical Simulation Results and Discussion

3.1. Effect of Coolant Temperature on the Internal Temperature Field in the Central Cross-Section of the ECR

In order to study the effect of different coolant temperatures on the internal temperature of the ECR, the coolant temperature range of 288–303 K and the rest of the working condition parameters are the initial values of the four working conditions for simulation analysis. The details of the working conditions are shown in

Table 1. Specific process parameters of each working condition.

	Injection Speed of Coolant (m/s)	Roll Speed (rad/s)	Coolant Temperature (K)	Initial RollTemperature (K)	Internal Heat Source Temperature (K)
Working condition, a	5	12	288	303	373
Working condition, b	5	12	293	303	373
Working condition, c	5	12	298	303	373
Working condition, d	5	12	303	303	373

. Due to the need to maintain the temperature field in a relatively stable state during the use of ECR, the external cold source and the internal heat source no longer affect the temperature field inside the roll, sufficient to drive the large-scale thermal force profile of the roll. Therefore, that moment was used as the object of analysis in this study to determine the effectiveness of that cooling strategy. The results of numerous simulations indicate that the working conditions in this study have reached a relatively steady state at 120 s. In order to facilitate a more detailed analysis of the internal temperature field of the ECR, in this study, the region where the internal temperature of the ECR was higher than the initial temperature was defined as the high-temperature influence zone of the internal heat source, and the region where the temperature was lower than the initial temperature was defined as the penetration influence zone of the external cooling mechanism.

Figure 6 shows the temperature distribution inside the roll in the central cross-section for different coolant temperatures. The results demonstrate a gradual decline in the roll temperature from the inner surface to the outer surface, exhibiting a distinct radial temperature gradient. This shows that the external cooling mechanism exerts a discernible influence on the ECR temperature field. This effect is predominantly observed on the ECR surface and within a limited radial inward distance. The built-in high-temperature heat source continues to provide a heat flow to the inner wall area of the roll, forcing the external cooling mechanism to work laboriously on the deeper areas of the roll. As illustrated in Figure 6a–d, it can be observed that the extent of the high-temperature influence zone of the internal heat source gradually increases with an increase in the coolant temperature. In contrast, the range of the penetration influence zone of the external cooling mechanism decreases. Figure 6e shows the quantitative pattern of the temperature gradient variation inside the ECR, and the results show that the radial temperature gradient can be considered in two stages. In the first stage, the temperature curve at four different T₁s is almost the same in the interval of radial distance of 0.075–0.12 m. This indicates that heat conduction from the heat source dominated in this range, and the rate of heat conduction inside the roll was much greater than the cooling rate. In the second stage, a clear differentiation in the temperature curve was observed in the radial distance greater than the 0.12–0.2 m interval. The lower the T₁, the steeper the radial temperature curve, indicating that a lower T₁ increases the heat dissipation rate and improves the cooling effect. The value of T₁ from the 288 K to 303 K interval changes in the radial distance of 0.2 m, the temperatures were 288.93 K, 293.76 K, 298.43 K, and 303.19 K, and the ECR surface temperatures were close to the T₁. The locally enlarged graph in Figure 6e demonstrates that h gradually increases with the decrease in T₁. At the value of T₁, which is 288 K, h reaches a maximum value of 0.05 m. The above analysis shows that using a lower T₁ can more effectively increase the depth of h and reduce the internal temperature of the ECR, which has a better cooling performance.

The analysis of the circumferential temperature uniformity of the ECR surface is different, and the temperature distribution of the ECR surface at different coolant temperatures is shown in Figure 7. The results show a non-uniform distribution of ECR surface temperatures under different operating conditions. However, as the T₁ increases from 288 K to 303 K, the temperature fluctuation on the ECR surface decreases significantly, and the temperature distribution tends to be uniform. This indicates that at higher T₁s, the temperature control of the ECR surface is more effective, and the heat distribution is more balanced. The values of T₁ were 288 K, 293 K, and 303 K when the shape of the ECR surface temperature distribution was similar to a bow and 298 K when the shape of the distribution was similar to a gourd shape. The T₁s of 288 K, 293 K, and 298 K in the ECR surface were 85° to 170° (i.e., the lower right surface) and the interval temperature was larger. This is due to the roll in the process of rotational movement leading to the formation of coolant on the surface of the ECR film, which in turn affects the cooling effect of the local area. The lower the T₁, the greater the peak of the temperature difference; due to the low-temperature conditions, the coolant may be rapidly evaporated on the surface of the ECR, resulting in heat that cannot be dissipated in time. At the value of T₁ which is 303 K, the roll indicates a greater degree of uniformity, with no significant temperature differential observed on the lower surface. This suggests that a high T₁ is more effective in preventing the formation of temperature gradients in this region.

To investigate the uniformity of the circumferential temperature distribution on the ECR surface further, the distribution of temperature uniformity on the ECR surface under the influence of different coolant temperatures is presented in Figure 8, where Figure 8a shows the value of the extreme difference of ε as a function of time for different T₁s. The results demonstrate that the slope of the value of ε increases with a reduction in T₁ during the heat transfer period of 0–80 s. Furthermore, the maximum value of ε is 0.76 K at the initial moment of 10 s and a T₁ of 288 K. It is because the lower T₁ considerably increases the intensity of convective heat transfer at the fluid–solid coupling surface, resulting in a more intense heat transfer process and therefore more significant temperature fluctuations. The lower value of T₁ results in a considerable intensification of the convective heat transfer at the fluid–solid coupling surface, leading to a more intense heat transfer process and, consequently, greater temperature fluctuations. With the increase in cooling time, the ε of the circumferential temperature of the ECR surface under the four conditions is gradually decreased. This is because with the rotating roll, all parts of the ECR surface are in direct contact with the coolant, increasing the intensity of the convection heat transfer. Consequently, the ε is gradually reduced, and in approximately 80 s, it reaches a stable state. Under the four working conditions, at the final moment of 120 s, the T₁s ranked from low to high ε were 0.21 K, 0.16 K, 0.11 K, and 0.07 K. This indicates that the temperature differential between the ECR surface and the inner area gradually diminishes over time. The immediate impact of the coolant on temperature becomes less pronounced.

Figure 8b shows the γ versus time for different coolant temperatures. The results show a rapid decrease in the average circumferential temperature gradient γ during 0–60 s at the onset of cooling. Applying a lower T₁ (e.g., 288 K) resulted in larger initial gradients, with a maximum γ value of 0.012 K/° at 10 s. In comparison, the initial gradient at the value of T₁ is 303 K, which was markedly smaller, with a γ value of 0.005 K/°. The rate of decline of γ slows down beyond 60 s and gradually stabilizes over the next few hours. The rate of γ varied at different T₁s, with the coolant at lower temperatures causing γ to fall more rapidly. At 120 s, the values of γ were 0.0035 K/°, 0.003 K/°, 0.0025 K/°, and 0.0018 K/° for the four working conditions, which indicated that the roll gradually reached a thermal equilibrium state, implying that the heat was balanced between the conduction inside the roll and the cooling outside. In conclusion, it can be seen that the use of lower T₁s increases the values of ε as well as γ, making the temperature distribution inhomogeneity on the ECR surface increase.

3.2. Effect of Initial Roll Temperature on the Internal Temperature Field in the Central Cross-Section of the ECR

Figure 9 shows the temperature distribution inside the roll in the central cross-section for different initial roll temperatures. As illustrated in Figure 9a–d, the temperature distribution within the roll is markedly influenced by the initial temperature of the roll. With an increase in the T₂, the range of the high-temperature influence zone of the internal heat source gradually expands. In contrast, the range of influence of the external cooling mechanism diminishes. This phenomenon indicates that an elevated T₂ intensifies the heat accumulation effect of the internal heat source, leading to a more pronounced heat accumulation in the roll center. Figure 9e illustrates the quantitative law governing the change in temperature gradient within the ECR. The results demonstrate that each roll’s radial temperature curve exhibits a consistent temperature decline from the center to the outside. The higher the T₂, the slower the radial temperature curve decreases. This shows that the cooling effect of the coolant becomes more pronounced as the radial distance gradually reaches the surface position. The comparative analysis of the ECR surface in the 303 K to 333 K h value reveals a notable correlation between the initial temperature of the roll and the resulting h value. As the temperature decreases, the h value also decreases, exhibiting a discernible trend. The observed range of h values, from lowest to highest, is 0.03 m, 0.05 m, 0.056 m, and 0.063 m. This phenomenon can be attributed to the elevated T₂, which gives rise to a heightened temperature differential between the ECR surface and the coolant. This, in turn, precipitates an intensification of the surface convective heat transfer, resulting in a rapid loss of heat from the surface. Meanwhile, the internal heat conduction rate is relatively sluggish because the heat is not dispersed outward promptly, so the h is shown to be larger. The above analysis shows that a higher T₂ can increase the depth of h more effectively, and the cooling performance is also better.

The analysis of the circumferential temperature uniformity of the ECR surface is different, and the temperature distribution of the ECR surface at different initial roll temperatures is shown in Figure 10. The results show that the effects of different T₂s on the circumferential temperature of the ECR surface show a similar trend; the shapes are approximately circular, and the temperature changes are regular and symmetrical as the angle changes. As the T₂s increase from 303 K to 333 K, the temperature fluctuation on the ECR surface increases significantly and reaches its maximum value in the region around 90° and 310° on the ECR surface. The data demonstrate that an elevated T₂ enhances the temperature disparity between the fluid–solid coupling surfaces, thereby augmenting the convective heat transfer intensity and engendering a more disparate temperature distribution due to the inability of the higher temperature coolant to establish a stable cooling layer on the ECR surface.

In order to investigate the uniformity of circumferential temperature distribution on the ECR surface further, the distribution of ECR surface temperature uniformity under the influence of different initial roll temperatures is shown in Figure 11. Figure 11a shows the variation in ε with time, and Figure 11b shows the variation in γ with time for different temperature conditions. Overall, both curves can be viewed in two stages. In the initial stage (0–80 s), the temperature changes occur faster in all four working conditions, and with the increase in the ECR surface temperature, ε and γ have more changes. After a cooling time of 80s, both ε and γ tend to level off, which indicates that the ECR surface’s temperature uniformity becomes more uniform with the increase in the cooling time. It was further found that when the value of T₂ is 303 K, the temperature polarity was consistently low, with an initial value of ε of 0.24 K, which eventually converged to 0.11 K, and a temperature gradient with an initial value of γ of 0.005 K/°, which ultimately converged to 0.0025 K/°. The results demonstrate that the coolant can effectively cover the ECR surface, ensuring superior cooling uniformity at T₂ lower. At the value of T₂ of 333 K, the value of ε is 1.74 K, eventually converging to about 0.6 K. The value of γ is 0.032 K/°, eventually converging to about 0.0068 K/°. At this time, both ε and γ reach the highest value, indicating that T₂ exacerbates the intensity of convective heat transfer, resulting in a significant increase in temperature distribution inhomogeneity.

3.3. Effect of Internal Heat Source Temperature on the Internal Temperature Field in the Central Cross-Section of the ECR

Figure 12 shows the temperature distribution inside the roll in the central cross-section for different internal heat source temperatures. The results show that with the T₃, the radial temperature gradient of the roll becomes more significant, and the high-temperature influence area of the internal heat source gradually increases. The penetration influence area of the external cooling mechanism becomes smaller. This is because the higher the T₃, the faster the rate of internal heat conduction to the outside, resulting in a significant increase in the temperature of the central region of the roll; a higher heat source temperature increases the thermal energy inside the roll, making the radial temperature gradient greater. It was additionally determined that h diminishes gradually as the temperature of the T₃. The values of h were found to be 0.0298 m, 0.0238 m, 0.0179 m, and 0.0159 m for the four operating conditions of the T₃, ranging from 373 K to 523 K in that order. This phenomenon can be attributed to heat conduction, which transfers heat from the center to the edge. The high-temperature region demonstrates a higher heat conduction efficiency and a relative lack of external cooling effects. This leads to a reduction in the efficiency of the cooling process within the roll at higher T₃ values. The above analysis demonstrates that an elevated T₃ value impedes the transfer of external cooling mechanisms to the interior of the roll, leading to a notable decline in the h value. Consequently, the efficacy of the cooling process is diminished.

Figure 13 shows the surface temperature distribution of the ECR for different T₃ values. The findings indicate that the impact of varying T₃ values on the circumferential temperature of the ECR surface is analogous to the differing T₂ values. Moreover, the shapes are all approximately circular, demonstrating a robust correlation. As the T₃ value was increased from 373 K to 523 K, there was a notable rise in the temperature fluctuation on the ECR surface. Furthermore, this trend remained consistent with the regions affected by different T₂ values. This demonstrates that an increase in either T₃ or T₂ results in a more significant generation of internal heat, thereby accelerating the heat transfer rate and leading to a rise in the inhomogeneity of the temperature distribution on the surface of the ECR.

In order to further investigate the uniformity of the circumferential temperature distribution on the ECR surface, the distribution of the ECR surface temperature uniformity under the influence of different internal heat source temperatures is shown in Figure 14. Figure 14a shows the variation in ε with time, and Figure 14b shows the variation in γ with time for different internal heat source temperature conditions. The data indicate that, over time, both ε and γ in each T₃ condition can be divided into three phases. In the initial homodyne stage (0–40 s), the rate of decline of the ε curve is essentially consistent across the four working conditions. However, there is a notable divergence in the γ curve, with a distinct difference observed when the value of T₃ is 423 K. The remaining conditions exhibit a high degree of similarity. This suggests that during the initial cooling phase, the convective cooling effect predominantly influences the ECR surface temperature and that the insufficient heat transfer rate does not directly affect the ECR surface. In the second descending phase (40–80 s), the decline in the ε and γ curves decelerated in conjunction with an increase in T₃. The transfer of heat from the interior of the ECR to the exterior via heat conduction results in a gradual reduction in the cooling effect of the coolant. The extent of this effect is contingent upon the T₃ value, with higher T₃ values exhibiting a more pronounced impact. Subsequently, the ECR surface temperature uniformity was optimal at approximately 80 s, with ε values of 0.0721 K, 0.0801 K, 0.0879 K, and 0.0958 K. During the initial phase of the process (80–120 s), an elevated T₃ value corresponds to a more precipitous rise in ε and γ curves. This suggests that the temperature of the coolant increases gradually over time, accompanied by a concurrent decline in its cooling efficiency. Concurrently, the heat generated within the ECR is continually conveyed to the surface, exacerbating the inhomogeneity of surface temperatures. From the above analysis, it can be seen that higher T₃ leads to a significant increase in the ε and γ values at the later stages of cooling, decreasing the temperature uniformity of the ECR surface.

3.4. Effect of Roll Speed on the Internal Temperature Field in the Central Cross-Section of the ECR

Figure 15 shows the temperature distribution inside the roll in the central cross-section for different roll speeds. The findings indicate that the temperature field within the ECR exhibits a gradual decline from the center towards the periphery under all w process parameters, demonstrating a distinct radial temperature gradient. When the final temperature field tends to stabilize, the value of w within a reasonable range of technology to apply does not significantly affect the internal temperature distribution, the radial temperature distribution curves almost overlap, and the values of h are all 0.0298 m. This phenomenon can be attributed to the transfer of heat from the center to the surface, which occurs primarily through thermal conductivity. In contrast, w exerts a predominant influence on convective heat transfer at the surface, with a comparatively lesser impact on the internal radial temperature distribution.

Figure 16 shows the surface temperature distribution of the ECR for different roll speeds. The results show a non-uniform distribution of ECR surface temperatures under different operating conditions. However, the regions exhibiting significant temperature fluctuations exhibit notable differences. At a w value of 8 rad/s, the temperature distribution assumes a shape similar to a Y shape. Furthermore, the surface temperature of the ECR exhibits significant fluctuations, ranging from approximately 95° to 345°. At a w value of 20 rad/s, the shape of the temperature distribution is similar to a bow, and the ECR surface temperature fluctuates the most in the interval from 210° to 320°. When the w value is 12 rad/s and 16 rad/s, the temperature distribution exhibits a shape analogous to that of a gourd. The range of temperature fluctuations on the ECR surface is primarily concentrated in the upper left surface of the roll.

In order to further investigate the uniformity of circumferential temperature distribution on the ECR surface, the distribution of ECR surface temperature uniformity under the influence of different roll speeds is shown in Figure 17, where Figure 17a shows the variation in ε with time, and Figure 17b shows the variation in γ with time for different roll speed conditions. As illustrated in Figure 17a, the value of w demonstrates a decline over time, followed by an upward trend. In the case of a velocity value of 20 rad/s and 10 s, the maximum value is 0.3692 K. In contrast, the remaining 10 s velocity conditions are approximately 0.22 K. This elucidates the phenomenon whereby an excessive value of w amplifies the intensity of convective heat transfer on the ECR surface, resulting in an augmented circumferential temperature inhomogeneity. At the final 120 s, the maximum value of ε is 0.1501 K for a w value of 8 rad/s, followed by a value of ε of 0.1409 K for a w value of 16 rad/s. This demonstrates that a low w value also increases the circumferential inhomogeneity of the ECR surface. As illustrated in Figure 17b, the γ curve exhibits a notable increase in slope with an increase in w. The γ values for the four conditions are observed to be 0.1467 K/° in the 80 s, and a smooth trend emerges. In conclusion, a lower w causes the γ value to flatten but increases the ε value during the final cooling stage. Conversely, a higher w leads to a tremendous rolled ε value at the early stage of the cold zone. Therefore, selecting an appropriate process interval (e.g., 12 rad/s) for controlling the w in order is crucial to achieving uniformity in the surface temperature distribution of the ECR during the entire cooling stage in actual production.

3.5. Effect of Coolant Injection Speed on the Internal Temperature Field in the Central Cross-Section of the ECR

Figure 18 shows the temperature distribution inside the roll in the central cross-section for different coolant injection speeds. The results show that the effect of different v on the internal temperature field of the ECR is similar to that of w, which is not significant within the range of values taken for the technical application. The radial temperature distribution curves almost overlap, except for the h value of 0.0278 m for the v value of 3 m/s, which is 0.0298 m for the rest of the working conditions.

Figure 19 shows the surface temperature distribution of the ECR for different coolant injection speeds. The results show that the influence interval of a different v on the surface temperature of the ECR is markedly disparate. The shape of the temperature distribution is analogous to the bow shape at v values of 3 m/s and 10 m/s, the shape of the temperature distribution is analogous to the gourd shape at the condition of 5 m/s, and the shape of the fusiform shape is analogous to the fusiform shape at the condition of 15 m/s. Among the four cases, when the value of v is 3 m/s, the cooling effect of the left half of the roll (180 to 360°) interval is optimal, resulting in lower temperatures. However, the temperature of the right half of the roll exhibits significant fluctuations, and the surface temperature of the ECR is higher than that of the other three cases, reaching 299.3 K. For values of v equal to 10 m/s and 15 m/s, the larger temperature fluctuation intervals are located on the lower right surface. Notably, the fluctuation intervals for the value of v equal to 15 m/s are more concentrated in the range from 135 to 165°.

In order to investigate the uniformity of circumferential temperature distribution on the ECR surface further, the distribution of ECR surface temperature uniformity under the influence of different coolant injection speeds is shown in Figure 20, where Figure 20a shows the variation in ε with time, and Figure 20b shows the variation in γ with time for different coolant injection speed conditions. Overall, the ε and γ curves for each v condition over time can be divided into two distinct phases. In the initial phase (0–80 s), the ε and γ curves for the v value of 5 m/s have exhibited a relatively smooth profile. Conversely, both too large and too small v values have increased ε or γ. During the interval of the steady-state-like phase (80–120 s), the maximum value of ε was 0.2212 K for a final v value of 3 m/s, and the minimum value of ε was 0.008 K for a v value of 15 m/s. However, the excessive jet velocity significantly increased the value γ after 100 s. In conclusion, an excessive or insufficient value of the v parameter leads to an increase in the value of ε in the initial period, which results in an inhomogeneity of the local thermal convexity of the ECR.

4. Conclusions

In this paper, numerical simulations and a discussion of the radial cooling capacity of ECR and the circumferential temperature uniformity distribution in the central cross-section of ECR are presented under five process parameters for different coolant temperatures, initial roll temperatures, internal heat source temperatures, roll speeds, and coolant injection speeds. The following conclusions were drawn:

A two-dimensional fluid–heat coupling VOF model was constructed to simulate and analyze the radial cooling capacity and circumferential temperature uniformity distribution of ECR under different process parameters. The results of the base case simulation and analysis are in close agreement with the experimental data, with an error of no more than 5% of the maximum temperature. Thus, the accuracy of the model has been verified.

In the ECR radial cooling capacity, the lower the coolant temperature, the higher the initial temperature of the roll; the lower the temperature of the internal heat source, the effect on the cooling capacity is more obvious, and the greater the depth of the cooling depth layer. Furthermore, the radial cooling capacity is not significantly influenced by variations in roll speed and coolant injection speed, except for surface temperature uniformity. To ensure optimal cooling capacity, improvements can be made in the following three key areas: the coolant temperature, the initial temperature of the roll, and the temperature of the internal heat source.

Regarding the ECR surface temperature uniformity, excessively low coolant temperatures, excessively high initial roll temperatures, and internal heat source temperatures contribute to increased surface temperature non-uniformity. An increase in the roll speed and coolant injection speed that is either too high or too low will result in an inhomogeneous temperature distribution on the ECR surface. It is, therefore, advisable to select a reasonable range to avoid this phenomenon. In order to mitigate the detrimental impact of the thermal contribution roll profile on the production of strips in actual production, it is essential to implement appropriate process parameters, such as a roll speed of 12 rad/s and a coolant injection speed of 5 m/s.