Online Roll Force Model for Non-Oriented Electrical Steel Considering Temperature-Transformation-Roll Force Coupling Effect

Abstract

During the finish rolling process of non-oriented electrical steel, the transformation from austenite to ferrite will occur. The uneven temperature along width and thickness direction will lead to the difference of phase transformation. Meanwhile, the latent heat accompanying transformation will affect the distribution of temperature. Due to the high-temperature deformation characteristic in dual-phase region is exactly opposite to that in single-phase region, the transformed fraction should be considered in the calculation of roll force. In order to make up for the deficiency of the traditional roll force prediction model. The transformation kinetics model, latent heat model and deformation resistance model for different phase regions which are established in our past work are adopted. The online prediction model of roll force of non-oriented electrical steel considering Temperature-Transformation-Roll force coupling effect are eventually obtained by taking the above basic models into the temperature model based on two-dimensional alternating difference method, which is able to calculate the distribution of temperature field and phase field and predict the roll force of each stand in different phase regions. After comparison and verification, the prediction errors of finish rolling temperature can be controlled within 10°C by new model proposed. The position of transformation is consistent with the abnormal wave observed. The proportion of rolled coils whose prediction error over 10% is decreased from 37.1% to 8.5% and the proportion of rolled coils whose prediction error less than 5% is increased from 15.8% to 41.2%. The computing time is only 68 ms which proves the feasibility of online application.

Fig. 2 The flow chart of the roll force prediction model.

1. Introduction

Because of the particularity of material composition and hot rolling schedule, the non-oriented electrical steel is more difficult than the plain carbon steel in the control of strip thickness and shape, which is known as the craft in the steel products.^1–4) One of the most difficulties in the practical production is the phase transformation from austenite to ferrite occurred in the hot finish rolling process. There will be uneven temperature and phase structure along strip width and thickness, and the coupling effects of temperature, phase transformation and roll force are needed to be fully considered, as shown in Fig. 1. Temperature will affect the deformation resistance of material and further contribute to the roll force. The heat produced by plastic deformation in the roll gap will raise temperature.⁵⁾ In addition to the above common coupling effects, the effect of phase transformation should be taken into account particularly. On the one hand, the latent heat is released during the phase transformation, which has a compensation effect on temperature and in turn affects the progress of phase transformation.⁶⁾ On the other hand, the mixed structure of austenite and ferrite caused by incomplete phase transformation shows the opposite high-temperature deformation characteristic from the single phase of austenite. In such case, the roll force is not only determined by the macro deformation conditions such as temperature, strain and strain rate, but also related to the phase structure.⁷⁾ At the same time, the deformation storage energy can induce the occurrence of phase transformation.⁸⁾ The production data indicate that the traditional austenitic rolling model is unable to quantify the effect of phase transformation on the roll force in the dual-phase region, so that the computational error of roll force, the thickness deviation and abnormal waves are common to occur.⁹⁾

Fig. 1

Coupling relationship among temperature, phase transformation and roll force.

To make up for the shortage of the existing models, the effect of phase transformation on the deformation resistance of non-oriented electrical steel are mainly focused on to improve the prediction accuracy of the roll force. C.S. Li et al.,¹⁰⁾ S.Z. Wang et al.¹¹⁾ and Y.D. Xiao et al.¹²⁾ adopted the traditional models to represent the deformation resistance of the austenite region, the ferrite region and the dual-phase region, which quantified the effects of deformation temperature, strain and strain rate on the deformation resistance in each phase region. However, such models are essentially statistical models, which adopt the functions of macro deformation factors and simply couple or equivalently deal with each effect, so that the effect of microstructure evolution on the macro mechanical deformation behavior cannot be reflected in the dual-phase region. To improve this situation, W.G. Li et al.¹³⁾ introduced the volume fraction of ferrite into the calculation of deformation resistance in the dual-phase region to represent the mixed state of the phase structures, and the expression used to obtain the transformed fraction is a simple cosine function that is related to the temperature change and independent with time. However, in fact, the transformed fraction varies with the cooling rate, which is determined by temperature and time simultaneously. In addition, the existing models only focus on the evolution law along the rolling direction and neglect the uneven temperature and phase structure along the strip width and thickness, which limit the prediction accuracy of roll force. Furthermore, the temperature compensation of latent heat should also be quantified. Therefore, mathematical models need to be established to represent the effect of phase transformation on the roll force.

In present study, the transformation kinetics model, latent heat model and the deformation resistance model of non-oriented electrical steel which are established in our past work are adopted. Then these models are incorporated into the hot finish rolling model, in which the two-dimensional alternating difference method is adopted to calculate the distribution of temperature and phase fraction at any time, meanwhile the SIMS formula is adopted to calculate the roll force of each stand. Finally, the roll force prediction model considering the coupling effect of temperature, phase transformation and roll force are verified to meet the requirement of online application in calculation accuracy and computation time for the industrial production of non-oriented electrical steel.

2. Temperature-Transformation-Roll Force Coupling Model

In order to avoid the heterogeneity of temperature and microstructure resulted from uneven cooling as far as possible, the water cooling system between the mill stands are usually closed during the finish rolling process of non-oriented electrical steel. Therefore, the whole rolling process can be divided into two kinds of heat exchange zones, which are the rolling deformation zone and the interstand air-cooling zone. Based on the two-dimensional alternating difference method, the roll force prediction model considering the multi-field coupling effect is established. The flow chart of the model is shown in Fig. 2.

Fig. 2

The flow chart of the roll force prediction model.

2.1 Basic models

The basic models used in the coupling calculation, such as transformation kinetics model, latent heat model and the deformation resistance model, are established by thermal expansion test, the DSC thermal analysis test and the hot compression experiment in our past work. So here is a brief description.

2.1.1 Transformation kinetics model

The strip in the hot finish rolling undergoes a continuous cooling process which can be quantified by a combination of the isothermal transformation kinetics model¹⁴⁾ and the Additivity Principle.¹⁵⁾ As shown in Ref. 16), the Avrami equation was employed to characterize the isothermal transformation kinetics as shown by eq. (1). By fitting the measured Dilatometer-Time data, the reaction rate k and the Avrami exponent n in eq. (1) were derived as eq. (2) and (3). Finally, in combination with the Additivity Principle, the transformed fraction in the continuous-cooling transformation process at any time t_s+1 was expressed by eq. (4) and (5).   

\begin{equation} X(t) = 1 - \exp (-kt^{n}) \end{equation} (1)

  

\begin{equation} k(T) = \exp (- 0.103T + 96.989) \end{equation} (2)

  

\begin{equation} n = 1.6 \end{equation} (3)

  

\begin{equation} X(t_{s + 1}) = 1 - \exp [- k_{s}(t_{s}^{v} + t_{s + 1} - t_{s})^{n_{s}}] \end{equation} (4)

  

\begin{equation} t_{s}^{v} = \root n_{s} \of {- \frac{\ln (1 - X(t_{s}))}{k_{s}}} \end{equation} (5)

where X is the transformed fraction. k is the reaction rate. n is the Avrami exponent. t is time, s. T is temperature, °C. t_s and t_s+1 are the start and end times of one of the short-time isothermal processes. t_s^v is the virtual time to achieve the fraction X(t_s) with the reaction rate k_s and Avrami exponent n_s at the temperature T_s.

2.1.2 Latent heat model

The latent heat is released simultaneously with the phase transformation, which will affect the temperature distribution of strip, and in turn affect the amount of transformation. As shown in Ref. 16), the latent heat flux is proportional to the transition rate, the formula of latent heat is defined as follow:   

\begin{equation} q = \rho H\frac{\partial X}{\partial t} \end{equation} (6)

where q is the latent heat flux due to transformation; ρ is density; H is specific enthalpy, 25.25 J/g; $\frac{\partial X}{\partial t}$ is the transition rate.

2.1.3 Deformation resistance model

The distinct phase structures in austenite, ferrite and dual-phase regions will correspond to different deformation resistance model. As shown in Ref. 17), the deformation resistance models for three phase regions are shown from eq. (7) to (9).   

\begin{equation} \left\{ \begin{array}{l} \sigma_{\text{A}} = ((\alpha_{\text{A}}\mu b)^{2}h_{\text{A}}\dot{\varepsilon}/s_{\text{A}}(1 - \mathrm{e}^{- s_{\text{A}}\varepsilon/\dot{\varepsilon}}) + (\sigma_{0}^{\text{A}})^{2}\mathrm{e}^{- s_{\text{A}}\varepsilon/\dot{\varepsilon}})^{\frac{1}{2}}\\ \alpha_{\text{A}} = 1.979\\ \sigma_{0}^{\text{A}}(T) = - 0.098T + 144.467\\ h_{\text{A}}(T) = - 0.465T + 600.685\\ s_{\text{A}}(T) = 0.055T + 8.778 \end{array} \right. \end{equation} (7)

  

\begin{equation} \left\{ \begin{array}{l} \sigma_{\text{F}} = ((\alpha_{\text{F}}\mu b)^{2}h_{\text{F}}\dot{\varepsilon}/s_{\text{F}}(1 - \mathrm{e}^{- s_{\text{F}}\varepsilon/\dot{\varepsilon}}) + (\sigma_{0}^{\text{F}})^{2}\mathrm{e}^{- s_{\text{F}}\varepsilon/\dot{\varepsilon}})^{\frac{1}{2}}\\ \alpha_{\text{F}} = 1.993\\ \sigma_{0}^{\text{F}}(T) = - 0.287T + 306.487\\ h_{\text{F}}(T) = - 1.072T + 1053.492\\ s_{\text{F}}(T) = 0.036T + 91.867 \end{array} \right. \end{equation} (8)

  

\begin{equation} \sigma = \sigma_{A}(1 - X) + \sigma_{F}X \end{equation} (9)

where α is the material constant, μ is the modulus of rigidity, and b is the Burgers vector, σ₀ is the yield stress, σ is the deformation resistance, h is the hardening coefficient representing growth rate of dislocation density, s is the softening coefficient representing decreasing rate of dislocation density. X is the transformed fraction of ferrite. Subscript A and F denote austenite and ferrite, respectively.

2.2 Fundamental equations and boundary conditions of heat transfer

According to the Fourier law, the differential form of the heat balance equation of the strip can be written as follow:   

\begin{equation} c\rho \frac{\partial T}{\partial t} = \lambda \left(\frac{\partial^{2}T}{\partial x^{2}} + \frac{\partial^{2}T}{\partial y^{2}} \right) + q_{\textit{in}} \end{equation} (10)

where c is the specific heat capacity; ρ is the density; λ is the thermal conductivity; T is the strip temperature; x is the coordinate along the strip width; y is the coordinate along the strip thickness; q_in is the heat flux of inner heat source, and here is the latent heat flux.

The heat exchange forms in the rolling deformation zone are as follows: the heat transfer between strip and work rolls, the heat generation from plastic deformation and friction, the latent heat if phase transformation occurs. The heat exchange forms in the interstand air-cooling zone are as follows: the heat convection between the strip surface and the surrounding air, the heat radiation from the strip surface, the latent heat if phase transformation occurs. The heat transfer coefficients refer to the Ref. 18). In addition, due to the recrystallization and recovery have a significant elimination effect on the deformation storage energy, the inducing effect of deformation on phase transformation is neglected.

2.3 Temperature model based on the two-dimensional alternating difference method

From the existing researches, the finite difference method is generally used to calculate the temperature field of hot rolled strip.^19,20) Particularly the two-dimensional alternating difference method can not only meet the requirement of online application in stability and solution speed, but also can be used to calculate the temperature distribution of the cross section of strip. Thus this method is adopted for modelling.

Firstly, the cross section of strip with the area of B × H is discretized to NB × NH rectangular meshes with the same width and thickness, as shown in Fig. 3. The upper and lower surfaces of strip are subjected to transfer heat with air or work rolls, while both sides of strip only lose heat to air. T_air, T_roll are the temperatures of ambient air and work rolls respectively. i, j are the node numbers along x direction and y direction respectively. Secondly, the governing equation and boundary conditions are discretized in the explicit-implicit alternating pattern. For the first half of a time step, the implicit difference method is used along width direction and the explicit difference method is used along thickness direction. And for the second half of the time step, the explicit difference method is used along width direction and the implicit difference method is used along thickness direction. For different heat exchange regions and different heat exchange types of nodes on the same cross-section, the equilibrium equations are formulated in differential form for each control volume. Finally, the equilibrium equations are written in matrix form and solved by the chasing method to obtain the temperature distribution in the cross section of strip at any time.

Fig. 3

Two-dimensional differential grid for the strip.

2.4 Coupled calculation of temperature and phase transformation

The transformation kinetics model and the latent heat model are embedded in the temperature model based on the two-dimensional alternating difference method. At the beginning of each iteration step, whether the phase transformation occurs or not is judged by the critical temperature determined by CCT experiments. If the critical condition is reached, the ferrite fraction and transition rate can be calculated by the transformation kinetics model, which can be used to calculate roll force and latent heat subsequently. The latent heat released is calculated by the latent heat model, which is treated as an internal heat source in the heat balance equation to reflect the effect of phase transformation on temperature. Conversely, the change of temperature will affect the generation of ferrite in the next iteration step. In this way, the coupled calculation of temperature and phase transformation is realized.

2.5 Roll force calculation considering phase fraction

The roll force is calculated by the classical SIMS formula²¹⁾ as follow:   

\begin{equation} F = l \cdot B \cdot \sigma \cdot Q_{p} \end{equation} (11)

where, F is the roll force; l is the contact arc length; B is the strip width; Q_p is the influence coefficient of stress state; σ is the deformation resistance of material, which is derived by the deformation resistance model considering phase structure.

The temperature and phase fraction for the calculation of deformation resistance are the average values of all nodes in the cross section of strip, which is more accurate than the traditional model in which the temperature and phase fraction are selected as the values of the central point that is unable to reflect the complete information of the whole cross section of strip. Finally, the roll force of each mill stand in various phase regions are derived.

3. Results and Validation

The temperature-phase transformation-roll force coupling model established in present study can export the results of temperature field and phase field of cross section of strip at any time. To verify the accuracy of the model, the CSP production line is selected due to the temperature distribution in the cross section before rolling is almost uniform and easily obtained from pyrometer between the tunnel heating furnace and the finishing stands. According to the common rolling process of non-oriented electrical steel, the initial temperature is set as 1080°C. The slab size is 73.80 mm × 1250 mm. The ambient temperature is 25°C. The rolling parameters of each stand are shown in Table 1. By substituting the practical process data to the proposed model, the transversal temperature distribution of upper surface of strip at exit of F7 mill stand is calculated and compared to the measured value, as shown in Fig. 4. It can be seen that the results are in good agreement and the errors are within 10°C.

Table 1 The rolling parameters of non-oriented electrical steel.

Fig. 4

Comparison of predicted and measured result of temperature distribution at F7 exit.

Furthermore, the evolution process of phase transformation can be represented by the average volume fraction of ferrite. It can be seen from Fig. 5 that phase transformation occurs between F3 and F4 mill stands and the average volume fraction of ferrite is increased from 13.2% to 72.1% rapidly, which is consistent with the location of observed abnormal waves in Ref. 6). It can be seen from the distribution of phase field at F4 mill stand in Fig. 6 that the ferrite fraction of the upper surface, lower surface and both edges of strip are larger than that of central part, where the starting time is earlier and the transforming speed is faster.

Fig. 5

Average volume fraction of ferrite at each stand.

Fig. 6

Distribution of ferrite fraction in the cross section of strip at F4 stand.

From the above, the new model can accurately predict the temperature field and phase field, which not only contributes to the control of the occurrence location and progress of phase transformation by designing reasonable process parameters, such as load distribution, finish rolling temperature and so on, but also contributes to the size control of internal microstructure, which is essential for magnetic property.

In order to verify the prediction accuracy of roll force, the production data of non-oriented electrical steel in a whole rolling unit are taken into the new model. It can be seen from Fig. 7 that the prediction accuracy of the new model is better than that of the traditional model. The coil proportion of which prediction error are more than 10% is decreased from 37.1% to 8.5%, and the coil proportion of which prediction error are less than 5% is increased from 15.8% to 41.2%. In addition, the solution time for each coil is only 68 ms, which is suitable for online application.

Fig. 7

Comparison of prediction accuracy between traditional model and new model.

4. Conclusion

1. (1)    The temperature-phase transformation-roll force coupling model is established by incorporating the effect of phase transformation into the calculation of temperature and roll force. The model can not only obtain the distribution of temperature field and phase field at any time during hot finish rolling, but also accurately predict roll force in different phase regions, especially in the dual-phase region. It can also reflect the influence of non-uniform distribution of temperature and phase structure along width and thickness direction on roll force. It can make up for the deficiency of traditional prediction model of roll force in the production of hot-rolled non-oriented electrical steel.
2. (2)    The prediction accuracy of proposed model is verified by comparison. The finish rolling temperature errors along strip width are within 10°C. The position of transformation is consistent with the abnormal wave observed.
3. (3)    The coil proportion of which prediction error are more than 10% is decreased from 37.1% to 8.5%, and the coil proportion of which prediction error are less than 5% is increased from 15.8% to 41.2%. In addition, the solution time is only 68 ms, which is suitable for online application.

Acknowledgements

The authors would like to thank the National Natural Science Foundation of China (Nos. 51674028) and the Fundamental Research Funds for the Central Universities (FRF-TP-18-105A1 and FRF-NP-18-006) for supporting this research.

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