Keywords

1 Introduction

The steel industry is taking a huge leap towards green transition and fossil free steels during the next decade. This requires significant investments in new technologies and processes. Modeling plays a substantial role in this field. Specifically virtual rolling models have an important part in this transition. Authors have developed a virtual rolling model to obtain calculated thermal, mechanical and metallurgical state of the steel strip throughout the rolling process. The model uses detailed line layout which is required for correct positioning of boundary conditions, e.g. descalers, rolling stands, coil box, water cooling units, etc. Process and product data is needed to define parameters like alloying and dimensions of the slab, pass schedule, water flows among other predefined process parameters. The calculation begins when a slab is discharged from the walking beam furnace, see Fig. 1. Next the slab is transferred to descaling and after this to roughing process. The following process steps are coil boxing, finishing rolling and accelerated water cooling. The slab position on the production line is controlled according to process data, and all boundary conditions are affecting at a correct position in layout. Figure 1 presents the temperature development throughout the rolling process for the top and the bottom surfaces and center thickness of the strip. On an x-axis is the line layout. The thickness reduction of the strip over the process is also depicted in Fig. 1 with black dotted line.

Fig. 1
figure 1

Calculated temperature development of top and bottom surfaces and center thickness of the strip in virtual rolling model. Calculated temperature curves are compared to measured pyrometer values from industrial process

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In this investigation we have focused on one feature of our virtual rolling model, a simplified sub-model to define development of surface temperature of the work roll (WR) in hot strip rolling process and how it affects the temperature development of the strip in virtual rolling model.

The most crucial parameter of the virtual rolling model is temperature. Temperature of the strip is coupled with all phenomena occurring in hot strip rolling and miscalculation of temperature weakens reliability of calculated metallurgical and mechanical phenomena. The virtual rolling model considers continuously changing conditions of manufacturing process by reading the process data. However, there are also parameters that are not measured in the process data, e.g. surface temperature of the WR. Surface of the WR is subjected to sequential heating and cooling cycles with the net result of positive heat influx [1]. In industrial process, this may lead to significant thermal expansion of the work rolls affecting the roll bite geometry and further the flatness of the strip and wearing conditions [2, 3]. The developed virtual rolling model is 1-dimensional and only changes in thermal conductance between the strip and WR are considered. However, alternating conditions in thermal contact conductance along the produced strip must be calculated in virtual rolling model. Several studies on the topic focus on heating and cooling cycles during one WR revolution [1, 4, 5], while virtual rolling model must calculate how the WR surface temperature evolves over the full length strip rolling as studied in [6]. The number of the WRs in virtual rolling model is 14 and surface temperature must be updated for all of those at every time step which set limitations for computing time of this individual sub-model. The virtual rolling model must be fast to be used as an online calculation tool in industrial systems and thus the simplified model for calculation of WR surface temperature is developed. The model is based on results of detailed Finite Element (FE)-model presented in this paper.

2 Configuration and Results of FE-Model

FE-model for six stands finishing mill is presented by authors in previously published articles [7, 8]. The modeling principles, physics and mechanics of this model are based on the same articles. The model utilizes non-linear facilities of a coupled temperature-displacement solver of the software package Abaqus™ Explicit. Finely discretized geometry is meshed with plane strain CPE4RT thermally coupled quadrilateral elements with enhanced hourglass control. Eight elements were used over the strip thickness and element length in rolling direction was 5 mm in the strip and on the contact surface of the WR. Exceptionally, 2 mm element length in rolling and 4 mm in vertical direction was used for the simulation 2 in Table 1 to ensure proper contact between the strip and the WR. FE-model contains automatic roll gap control which calculate the contact pressure and friction shear stress distributions in roll bite to predict roll force. Setup calculations and roll gap control are introduced by authors in [9]. The FE-model of the finishing mill is modified as a single stand model to simulate heat transfer during 75 m long transfer strip rolling on the first rolling stand of the finishing mill. Single stand FE-model is shown in Fig. 2.

Table 1 DOE for WR temperature evolution in single pass FE-simulation
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Fig. 2
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FE-model of the single stand transfer bar rolling

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2.1 Thermal Boundary Conditions of WR

Surface of the WR goes through sequential heating and cooling cycles during rolling pass. To obtain reliable surface temperature development, all thermal boundary conditions affecting the WR must be considered. The main heat transfer mechanisms in this case are heat transfer by conduction from surface of the WR to inner of the WR, thermal contact conductance between the strip and the WR, heat radiation and convection to ambient and between the strip and WR when there is no mechanical contact. And the most efficient cooling mechanism is water cooling nozzles.

The heat transfer efficiency of water cooling highly depends on the temperature of steel surface, see Fig. 3 (cooling water is assumed to be near room temperature, 20 °C). At high surface temperature of steel, a steam film (Leidenfrost effect) forms and isolating steam layer between the surface of the steel and liquid water weakens the heat transfer [10]. Because effective heat transfer coefficient (EHTC) is significantly dependent on surface temperature of WR, it must be considered in the FE-model. EHTC on the surface of WR due to water cooling spray is defined by Eq. 1, used by Martin in [10] and originally experimentally determined in [11].

$${\omega }_{c}=3.16\cdot {10}^{9}{\zeta }_{0}{\zeta }_{1}{\left[\left(u-{u}_{0}\right)-{\zeta }_{2}\left(u-{u}_{3}\right)\right]}^{-2.455}{\dot{\theta }}^{0.616},$$
(1)

where \({\zeta }_{0}\), \({\zeta }_{1}\) and \({\zeta }_{2}\) are defined in the following Eqs. 2, 3 and 4:

$${\zeta }_{0}=\left\{\begin{array}{c}\begin{array}{cc}0 & u<{u}_{0}\end{array}\\ \begin{array}{cc}\frac{u-{u}_{0}}{{u}_{1}-{u}_{0}}& u\in \left[{u}_{0}, {u}_{1}\right],\end{array}\\ \begin{array}{cc}1 & u>{u}_{1}\end{array}\end{array}\right.$$
(2)
$${\zeta }_{1}=\left(1-\frac{1}{1+{\text{exp}}\left(\frac{u-{u}_{2}}{40}\right)}\right),$$
(3)
$${\zeta }_{2}=\left(1-\frac{1}{1+{\text{exp}}\left(\frac{u-{u}_{3}}{10}\right)}\right)$$
(4)

with u0 = 273 K, u1 = 425 K, u2 = 573 K, u3 = 973 K, u is the node temperature of the WR surface and \(\dot{\theta }\) is the water flux (l/(min·m2)). It must be noticed that different water nozzle geometries and used water pressure may affect the EHTC on different systems and thus parameters of equations may need readjustment. EHTC is implemented into FE-model using VDFLUX-subroutine. In the VDFLUX, user must determine heat flux q on the picked surface and thus EHTC must be multiplied with temperature difference between cooling water and WR surface, \(q={\omega }_{c}\cdot ({T}_{WR}-{T}_{{\text{water}}})\). The cooling water temperature was set to 20 °C. Water flux \(\dot{\theta }=1000 l/({\text{min}}\cdot {{\text{m}}}^{2})\) in Fig. 3.

Fig. 3
figure 3

Effective heat transfer coefficient values for forced convection as function of WR surface temperature

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Schematic Fig. 4 shows assembly of cooling nozzles used in FE-model. Figure 4 shows simulated temperature history of WR surface node during one revolution. Numbers in the schematic roll Fig. 4 show the thermal boundary conditions of WR surface and the same numbering in right side chart shows the temperature history corresponding to the boundary conditions. WR and the strip are assumed to be initially isothermal, and temperatures are predefined as 45 and 900 °C at the beginning of simulation, respectively. In the first phase (Fig. 4), the contact with hot strip heats the WR surface at 198.6 °C. Magnitudes of convection, radiation and thermal conductance are applied as experimentally determined coefficients in [12]. The second phase contains heat conduction below the WR surface and heat radiation from strip to WR surface. Conductance in WR transfers heat efficiently from surface, simultaneously cooling the WR surface. Heat radiation has a minor effect compared to conductance. Next three phases consist of WR water cooling nozzles. The water impingement areas of the WR cooling nozzles are determined based on areas limited by cartesian coordinates due to revolving WR. The cooling efficiency of these units depends highly on water flux and pressure as well as temperature gradient between cooling water and WR surface. In addition, the heat conduction inside the WR decreases during the rolling period, and the WR surface achieves balanced heating and cooling cycles depending on pass schedule (Fig. 5). During phase six heat conduction cools the WR surface and WR is in contact with back-up roll (BUR). Heat conductance between WR and BUR can be assumed to be insignificant [1]. Thus, the contact is ignored due to relatively short contact length which would have required very fine discretization on contact surfaces, increasing computing time significantly. The last two phases of revolution include two cooling nozzles before next contact with strip surface. Surface temperature of the WR increases after phase eight due to thermal conductance from inside the WR to the surface.

Fig. 4
figure 4

Schematic figure of WR thermal boundary conditions and modeled temperature history of WR surface during one revolution (Sim. 3, Table 1)

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Fig. 5
figure 5

WR surface temperatures of simulations 3, 4 and 5 with different circumferential velocities of WR

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2.2 FEM Simulations and Results

Heat transfer phenomena in single pass transfer strip rolling were simulated in 7 different cases. The entry thickness of the transfer strip was set to 39.59 mm. Simulations were designed using three different reductions, circumferential velocities of WR and initial temperatures of transfer strips, see Table 1.

FE-simulations presented in Table 1 are shown in Figs. 5 and 6. Figure 5 shows the effect of rolling speed on the heating and cooling cycles. Only circumferential velocity of the WR was deviated between simulations 3, 4 and 5. Simulation 3 was run with the slowest rolling speed and long contact times with the strip and water cooling sprays cause high temperature variations on the WR surface. Simulation 1 requires the longest time to reach state of equilibrium between sequential revolutions. In equilibrium state, the lowest temperature during a single revolution is 148 °C after phase six in Fig. 4 and highest temperature 278 °C right after strip contact. When rolling speed increases in simulations 4 and 5, the state of equilibrium is reached faster as the same amount of revolution is achieved in a shorter time. Temperature differences during a single revolution are also significantly smaller. However, the lowest equilibrium temperature 148 °C after every revolution is approximately the same for all three simulations 3, 4 and 5. So the rolling speed has an effect only on time when the equilibrium state is achieved. It is also worth noting that idling revolutions after the rolling pass cools the surface of WR relatively fast regardless of rolling speed.

Fig. 6
figure 6

WR surface temperatures of simulations a 2, 4 and 6 with different thickness reductions and b 1, 4 and 7 with different transfer bar temperatures

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The effect of different reductions and transfer strip surface temperatures on warming of WR surface is depicted in Fig. 5a and b. With a small 3% reduction in simulation 2, the WR warming is almost imperceptible and equilibrium temperature is around 50 °C. This results from small reduction and very short contact length with rolled strip when there is less time for heat transfer and because the surface of the WR remains relatively cold after strip contact the cooling boundary conditions can keep equilibrium temperature low. The difference is clear compared to the high 48% reduction in simulation 6 where equilibrium is found near 150 °C which is the same with all other simulations except simulation no. 2. Also, simulations 1, 4 and 7 achieve the state of equilibrium at around 150 °C.

Simulations 3 and 6 have the highest temperature variations during a single revolution of the WR at equilibrium state. This is explained by slow circumferential WR velocity 1 m/s which leads to long contact time with all boundary conditions in simulation 3 and high thickness reduction 48% which emphasizes the heat transfer in strip contact due to long contact time in simulation 6. In any case, these two simulations achieve the same equilibrium state temperature of 150 °C with other simulations excluding simulation 2. This address that the higher surface temperature of the WR intensifies the cooling effect of water according to Eq. 1 and nearly the same equilibrium temperature of 150 °C between the heating strip contact cycle and cooling cycles is achieved with different rolling parameters.

3 Work Roll Warming Model for the Fast Virtual Rolling Model

To utilize FEM results in the fast virtual rolling simulations, a model based on the FEM results was developed. The FEM results were approximated so that the time \({t}_{2}\), required to reach ‘equilibrium’. Equilibrium is assumed to be reached when the temperature at the end of contact is \({T}_{\text{high}}\) and cools to \({T}_{\text{low}}\) during one revolution before the beginning of next contact. The \({t}_{1}\) is the time required to reach 90% of these temperatures. These four parameters were collected from FEM results, see Fig. 7. It was assumed that the heating of work roll would depend on the following parameters: strip temperature, contact length, contact time and cooling time. A Python library called scikit-learn was used to create linear regression predictor objects that read those four parameters and outputs \({t}_{1}, {t}_{2}, {T}_{{\text{high}}}\) and \({T}_{{\text{low}}}\). The input and output parameters for the linear regression predictor object are listed in Table 2. \({T}_{{\text{strip}}}\) is the temperature of the strip before the roll contact, \({t}_{{\text{contact}}}\) is the time that a single point on work roll surface is in contact with strip during one revolution and \({t}_{{\text{cooling}}}\) is the time that the point on the surface of WR cools before it hits the surface of the strip again. \(R\) in Table 2 is the thickness reduction. Values are obtained from simulations presented in Table 1. Once all these parameters are known, the work roll surface temperature can be approximated so that it is \({T}_{{\text{beginning}}}\) (Eq. 5) in the beginning of a single contact and rises linearly during the contact so that it is \({T}_{{\text{end}}}\) (Eq. 6) at the end of that contact.

$${T}_{{\text{beginning}}}=\left\{ \begin{array}{c}{{T}_{0}+t/{t}_{1}(0.9T}_{{\text{low}}}-{T}_{0}), t\le {t}_{1}\\ {T}_{{\text{low}}}(0.9+0.1(t-{t}_{1})({t}_{2}-{t}_{1}), {t}_{1}<t<{t}_{2}\\ {T}_{{\text{low}}}, {t\ge t}_{2}\end{array}\right.$$
(5)
$${T}_{{\text{end}}}=\left\{ \begin{array}{c}{{T}_{0}+t/{t}_{1}(0.9T}_{{\text{high}}}-{T}_{0}), t\le {t}_{1}\\ {T}_{{\text{high}}}(0.9+0.1(t-{t}_{1})/({t}_{2}-{t}_{1}), {t}_{1}<t<{t}_{2}\\ {T}_{{\text{high}}}, {t\ge t}_{2}\end{array}\right.$$
(6)

where \(t\) is the time since the head of the strip entered the roll bite and \(T_{0}\) is the initial temperature of the WR surface.

Fig. 7
figure 7

Result of FE-model utilized to derivate simplified equations for predicting WR surface temperature in virtual rolling model

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Table 2 The input and output parameters for the linear regression predictor object
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After the rolling pass is finished, the work roll surface temperature drops following Newton’s law of cooling as in Eq. 7.

$$ T\left( t \right) = T_{0} + \left( {T_{\text{low}} - T_{0} } \right){\text{exp}}\left( {kt_{c} } \right) $$
(7)

where \(t_{c}\) is the time since the finishing of the last pass, and k is a coefficient that was fitted to be −0.3.

4 Results and Discussion

Developed Eqs. 5–7 for predicting WR surface temperature in roughing and finishing mill processes are implemented in the virtual rolling model. The WR warming model is utilized to calculate development of strip temperature in virtual rolling model. The same sub-model for predicting WR warming in virtual rolling model was implemented for all WRs in the model. The virtual rolling model considers rolling pass mechanics and all thermal boundary conditions on the production line as well as metallurgical phenomena. The focus in the following results is only on temperature model of the virtual rolling model. Figure 8 presents the influence of WR surface temperature on the temperature development of bottom surface of the strip. Figures 9 and 10 show two different rolling cases: coil boxed and passed-through which means that strip is finishing rolled directly after roughing passes without coil boxing in hot strip production. Calculated temperature history for strip head and tail in both cases is shown. Calculation points are located 10% lengthwise from both ends of the strip so that thermal conductance in lengthwise direction does not have effect on temperature in calculation point (1D-model). The model layout mimics hot strip mill of SSAB Raahe. In the calculated cases all features of virtual rolling mill have been used and modeled cases compared to measured process data. Calculated and measured finishing rolling temperature (FRT) are depicted in Figs. 8, 9 and 10. FRT is the bottom surface temperature of the strip after the last finishing mill rolling pass. This temperature is critical for planning suitable water cooling practices and thus accurate temperature prediction for the FRT is required in virtual rolling mill.

Fig. 8
figure 8

Effect of WR surface warming on temperature development of bottom surface of the strip. Simulation with constant WR surface temperature (20 °C) and simulation with WR warming model implemented in virtual rolling model are presented

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Fig. 9
figure 9

Calculated temperature history of strip head and tail (bottom surface and center of the strip) in coil boxed strip production

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Fig. 10
figure 10

Calculated temperature history of strip head and tail (bottom surface and center of the strip) in passed-through strip production

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Two simulations throughout the virtual rolling model were run and compared in Fig. 8. In the other simulation constant WR surface temperature of 20 °C was used and another one utilized a developed WR warming model in thermal contact between the WR and the strip. Both simulations were calculated for the tail part of the strip when WR surface is clearly heated due to long lasting strip contact. Results in Fig. 8 are depicted on the finishing mill area after coil boxing process in virtual rolling model. Both calculated temperature curves of bottom strip surface and the measured FRT are presented in Fig. 8. The difference between temperature paths shows how significant difference in temperature development of strip surface occurs without considering the WR warming in virtual rolling model. On the x-axis is used normalized process time starting from discharge from walking beam furnace and ending to coiling.

The simulated strip in Fig. 9 is produced utilizing coil boxing process. It can be seen how the tail of the strip spent a long time coiled in the coil box. There is a clear difference in finishing mill rolling starting temperature (RST) between strip head and tail. The RST is used in Figs. 9 and 10 to point out the bottom surface temperature of the strip before finishing mill descaling process. There is no measured value for RST from process data. According to calculated temperature paths in Fig. 9, the finishing mill process of the strip head starts at lower RST than rolling of the tail. And because the FRT of the head of the strip is also higher than the FRT of the tail which finishing rolling process is completed much later, there must have been adjustments of finishing rolling parameters between rolling the head and the tail of the strip. This is explained by deceleration of tandem mill from 4.7 to 3.7 m/s (6th stand rolling speed) between the strip head and tail. This influences the time at which boundary conditions affect the virtual rolling mill due to which the tail of the strip loses more thermal energy than head of the strip. All heat transfer mechanisms last for a longer time and less adiabatic heat is produced in the strip. Measured FRT values confirm the results.

The other case in Fig. 10 is produced without coil boxing process. Unlike in the coil boxed case the finishing mill has accelerated from 4.2 to 5.1 m/s between the strip head and tail, leading to slightly higher FRT at strip tail. Measured FRT values agree with the calculated temperatures. In both cases, the new model for WR surface warming was used and according to results it is suitable for predicting WR surface temperatures in virtual rolling model. Ignoring WR surface temperature variations during the rolling period inevitably leads to significant temperature errors in virtual rolling model.

Results prove how essential continuously changing boundary conditions are for the virtual rolling model which calculates the entire hot rolling process. To simulate temperature development throughout the hot strip production line, all thermal boundary conditions must be considered. Simulation results of virtual rolling model were compared to the measured FRT because there are generally several minutes of rolling process past at this point and it is one of the most critical temperatures in the process regarding the final properties of the strip. So, the model must be able to predict FRT reliably. According to simulation results, the WR warming model predicts reliably WR surface temperature throughout the rolling process. The WR warming model is only one parameter which is changing constantly in industrial process. For that reason, the presented results contained all features of virtual rolling model. These are virtual rolling automation, temperature model, calculation module for metallurgical phenomena and thermo-mechanical module for rolling passes. The virtual rolling automation mimics the industrial automation system controlling the strip position, velocity and changes of direction in roughing. The temperature module acknowledges all thermal phenomena and boundary conditions and delivers them to other modules during the simulation. The metallurgical module calculates recrystallization, grain growth and phase transformations which are further utilized in thermo-mechanical rolling module which in turn calculate phenomena in roll bite predicting roll force and WR flattening based on contact pressure and friction shear stress between the strip and WR.

As an afterthought the thickness reduction parameter for simulation 2 was poorly chosen because such a small reduction is not generally used in hot strip rolling process. On further investigation 6-stands FE-model of finishing mill can be used to study temperature development of hot strip and WR surfaces in multi-stand rolling process. Also multiple strips could be simulated sequentially to investigate evolution of equilibrium temperature over rolling periods.

5 Conclusion

Work roll heating and cooling cycles were modeled with detailed FE-model to find out what kind of temperature variations work roll surface goes through during the rolling pass. It was also unknown that the work roll surface would reach equilibrium state at some point of rolling full length strip. Thermal cycles and equilibrium state were established in results of FE-model and a simplified version for virtual rolling model was derived from the FE-results. In validation results the temperature development of strip in virtual rolling model was modeled. The correct FRT value was obtained for hot strip rolling cases which means that the temperature development throughout the hot strip rolling process was modeled correctly. Calculated FRT after finishing mill corresponded with the measured FRT value from industrial process. The validity of the temperature calculation between strip head and tail was compared to the measured FRT with good correlation.

It must be considered that every cooling system for WR surface cooling is unique and water flows, impingements areas and water pressure may differ significantly. Thus, parameters used for effective heat transfer equations in WR water cooling in this study may vary noticeably in validation to other cooling systems. This may result in a different equilibrium temperature on WR surface in steady state rolling.

The virtual rolling model also considers the most essential thermal boundary conditions and includes virtual rolling automation which both are required to calculate the temperature evolution of the whole hot strip rolling process reliably.