J. Cent. South Univ. Technol. (2010) 17: 795-800

DOI: 10.1007/s11771-010-0558-5

Rolling force prediction for strip casting using theoretical model and artificial intelligence

CAO Guang-ming(曹光明), LI Cheng-gang(李成刚), ZHOU Guo-ping(周国平),

LIU Zhen-yu(刘振宇), WU Di(吴迪), WANG Guo-dong(王国栋), LIU Xiang-hua(刘相华)

State Key Laboratory of Rolling and Automation, Northeastern University, Shenyang 110004, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2010

Abstract: Rolling force for strip casting of 1Cr17 ferritic stainless steel was predicted using theoretical model and artificial intelligence. Solution zone was classified into two parts by kiss point position during casting strip. Navier-Stokes equation in fluid mechanics and stream function were introduced to analyze the rheological property of liquid zone and mushy zone, and deduce the analytic equation of unit compression stress distribution. The traditional hot rolling model was still used in the solid zone. Neural networks based on feedforward training algorithm in Bayesian regularization were introduced to build model for kiss point position. The results show that calculation accuracy for verification data of 94.67% is in the range of ±7.0%, which indicates that the predicting accuracy of this model is very high.

Key words: kiss point; Navier-Stokes equation; rheological properties; Bayesian method; generalization capabilities

1 Introduction

Traditional steel strip production is fundamentally changed by twin roll strip casting, during which the molten liquid steel is cast directly into strip, omitting the processes such as continuous metal casting, re-heating, and hot rolling. It is suitable for the steel industry development tendency of low energy consumption, short procedure, high speed and low production cost. In the past twenty years, aggregate investment for developing strip casting technology is equal to 1%-2% of investment for technology development of steel industry in the world. It can be foreseen that the commercial application of this technique would exert important influence on metallurgical industry and play a revolutionary role in progress of metallurgical industry. This technique is widely applied in fine crystalline and homogenous casting strip of stainless steel and carbon steel [1-3].

Rolling force is an important dynamic parameter that ensures that strip casting proceed steadily, which does not only affect control accuracy for shape and thickness of casting strip, but also affect strip quality. Therefore, it is very important to control rolling force exactly. Deformation procedure during strip casting is considered to be equal during hot rolling, and rolling force for strip casting is calculated by using hot rolling formula and ignoring their differences. In recent years, with deep investigation of rheology characteristic and deformation theory for strip casting, a model for rolling force during casting strip was built, in which static balance differential equation for deformation procedure in casting and rolling zones was deduced by using slab method, and solid phase fraction was introduced to characterize the effect of solid metal on deformation resistance in the model for deformation resistance in casting zone. ZHU et al [4] solved differential equations for casting and rolling zones by using Runge-Kutta method analytically according to friction condition, respectively. The calculation model was built by using non-grid finite element coupling method, and rolling force distribution during rolling was analyzed. Rolling reached the maximum value in the middle of deformation zone and decreased gradually at the entrance and the exit [5]. But there were some limitations during calculation of rolling force by using empirical equation due to the uncertainty of factors such as the kiss point.

In this work, based on analysis of solidification and deformation mechanism in the molten pool, solution zones were devided into two parts by kiss point position during strip casting. The mathematical model of rolling force was established on the basis of viscous fluid mechanics in liquid zone and mushy zone, and the traditional hot rolling model was still used in the solid zone. The differential equation for kiss point position was deduced by using empirical formula, considering geometrical relationship in the molten pool. The     kiss point position was acquired by solving differential equation, however, which was very complex. An intelligent algorithm [6] through which the kiss point positions were calculated by using the inversed algorithm according to measured rolling data and the model for the kiss point were proposed by introducing neural networks based on training algorithm of feedforward in Bayesian regularization [7]. There were lots of studies on neural networks based on Bayesian regularization abroad. Neural network was applied to position tracing of users of mobile phone, adopting such standard training methods as anti-propagation algorithm, Elman algorithm and Levenberg-Marquard algorithm in network training, and it was found that network training results by Bayesian regularization were better than those by other methods [8]. Bayesian regularization was firstly applied to the calculation of the kiss point position, and model for rolling force in strip casting was built by combining with other theoretical models mentioned above in this work. Fig.1 shows schematic view of model structure.

2 Rolling force modeling in liquid zone and mushy zone

The inner of the metal underwent a process of liquid phase to solid phase, and the physical characteristic of melted metal in deformation zone was similar to that of viscous fluid although the surface in contact of metal was already solidified in liquid zone and mushy zone. Therefore, Navier-Stokes equation was introduced as the fundamental equation to discuss the rolling force.

         (1)

where p represents the hydraulic pressure, Pa; F represents the volume force, N/kg; ρ represents the density, kg/m³; v_i represents the speed in direction i, m/s;  represents the partial derivative of v_i and η represents the viscosity coefficient, Pa·s.

Meanwhile, stream function  was introduced into the velocity filed establishment of the deformation zone, and the analytic equation of average unit compression stress distribution was deduced.

2.1 Basic assumptions

Based on rheological property of casting rolling, some assumptions were given to get the approximate analytic solution.

(1) Deformation in these zones was plane deforma- tion.

(2) Inertia force and volume force were neglectable.

(3) The metal in these zones was viscous and in- compressible fluid.

(4) The metal in these zones was low viscous fluid, and stress-strain relationship was

                                (2)

where  represents stress deviator tensor, Pa; and  represents strain deviator tensor, s^-1.

According to the assumptions, the simplified representations of Navier-Stokes equation were

                   (3)

                               (4)

2.2 Establishment of velocity field

Curvilinear coordinates ξ₁ and ξ₂ were introduced in the establishment of velocity field, and the deducing process was available in Ref.[9].

                     (5)

2.3 Velocity boundary conditions

Fig.2 shows schematic view of liquid zone and mushy zone. From Fig.2, thickness h_x of any cross section in these zones can be described as

                     (6)

where h_k represents the thickness of kiss point, m; and R

                                   (7)

Fig.1 Schematic view of model structure

Fig.2 Schematic view of liquid zone and mushy zone

                           (8)

Velocity field was established by Eq.(6), which satisfied both continuous flow function and velocity boundary conditions and was feasible to establish the velocity field using stream function.

2.4 Analytic equation of average unit compression stress distribution

Consideringlet ＝0, substituting it to Eqs.(3) and (8) leads to

               (9)

where d represents the thickness of the solidification endpoint, m. In this work, Runge-Kutta algorithm was used for numerically solving Eq.(9).

2.5 Regression model of temperature field

Temperature field calculation was the precondition of rolling force calculation. Much work on flow, heat transfer and solidification in twin-roll strip casting process was done by researchers at home and abroad, and a lot of numerical simulations which exhibited high precision were made [10-12]. The temperature field was unmeasurable and the model calculation needed fast- response ability. Considering the symmetry, coupled turbulent flow and temperature field of the twin-roll casting strip process were simulated by a two-dimensional finite element method. The following time-averaged general formula was given to describe the governing equations in a two-dimensional system of coordinates:

 (10)

where C_Φ represents the coefficient of instant and convection item; Γ_Φ represents the coefficient of diffusion; S_Φ represents the heat source; u_k represents the momentum in direction k; and  represents the equation variable.

The temperature field was unmeasurable and the model calculation needed fast-response ability, so on-line application of the finite element method was restrained. In this work, a regression model of temperature (t) was given by using data obtained by finite element simulation.

    (11)

where t_c represents the pouring temperature,℃, and

                      (12)

L represents the molten level, m, and

    (13)

v represents the casting speed, m/s, and

      (14)

x represents the present position, m; p_k represents the position of kiss point, m; and c_i (i=1, 2, …, 22) represents the regression parameter. Fig.3 shows the comparison between temperatures calculated by model and that by Ansys simulation. From Fig.3, it can be seen that the model has high precision.

Fig.3 Comparison of calculated temperature with model and simulated one

2.6 Viscosity model

The expression of viscosity of molten metals in liquid zone was:

                       (15)

where C_gas is the gas constant, J/(mol·K); u_A is the viscosity at melt point, Pa·s; C₁ is the specific heat capacity of liquid metals, J/(kg·K); T₁ is the temperature of liquidus line, K; and T is the temperature of liquid metal, K.

The viscosity in mushy zone can be measured through experiments [13]. The experiment results show that the whole zone could be divided by critical solid fraction and viscosity at solid fraction stages has different varying tendency. The empirical formula was as follows:

                    (16)

where f_s is the solid fraction; f_cr is the critical solid fraction; and u_A, M_s, k₁ and k₂ are the empirical formula parameters.

2.7 Solid fraction calculation model

Scheil’s model was used to calculate the solid fraction [14].

            (17)

where T_s is the solidus temperature,℃; and k₀ is the  model parameter.

3 Casting rolling force modeling in solid zone

Because the metal was completely solidified and its deformation process was similar to hot rolling, the traditional hot rolling model was still used.

                               (18)

where P_s is the casting rolling force in solid zone, N; B is the width of strip, m;  is the horizontal projection of contact area, m; Q_P is the influential coefficient; and K is the deformation resistance, Pa.

4 Kiss point calculation model

In order to guarantee high precision of the model, the model parameters must be modified using the measured data and instant message, which was called model self-learning. In the twin-roll strip casting process some variables such as turbulent flow and temperature fields were unmeasured and disturbed by many interacting parameters, so, Bayesian regularization was applied to the training of feedforward neural networks to improve the accuracy of this model.

4.1 Network principle

Bayesian regularization was applied to the training of feedforward neural networks to improve their generalization capabilities. Coupling with the Occam’s razor theory, a penalty item, which could characterize network complexity, was introduced into the performance function to prevent the occurrence of overfitting. The performance function was as follows:

      (19)

where E_D is the conventional error;  is the a penalty item of network complexity; w_i is the weight; t_jk is the expected output; y_jk is the actual output; O is the number of output variables; N is the number of training data; G is the number of weight groups; W_g is the number of weight in group g; and α_g, β were the hyperparameters.

4.2 Hyperparameters and log evidence (M) calculation

Hyperparameters (α_g, β) and M were calculated as follows [15]:

                                (20)

                              (21)

                   (22)

                  (23)

where H is the Hessian matrix of performance function (19);  is the number of effective parameters; P is the number of middle layer; E_D is the error of prediction; and I_k is the identity matrix.

4.3 Network training procedures

(1) Initialize the hyperparameters (α_g, β) and weight of the network using random numbers.

(2) Levenberg-Marquardt equation [16] is applied to the training of feedforward neural networks and updating α_g and β.

(3) Calculate performance function and M.

(4) Repeat steps (2) and (3) until the performance function is converged.

4.4 Input of network

The inputs of network contained pouring temperature, roll gap, casting speed, molten level, and cooling water temperature. In this work, the kiss point positions that were calculated by the measured data through the inversed algorithm were used to train the networks. There were 1 000 groups of calculated data, 70% of which were used to train the network and 30% of which were used to verify the network. The experimental material was 1Cr17.

4.5 Network structure selection

Fig.4 shows that there is a redundancy link when the node of middle layer is 2.

The optimal structure of network could be optimized according to the maximization of M and minimization of SEP (the square root of the mean squared error of prediction). Fig.5 shows that with the increase of M, the SEP decreases quickly. Fig.6 is local amplification of Fig.5. When the node of middle layer is 3, the SEP will have centralized distribution and the least forecasting error. So, the optimal structure of network has 3 nodes in the middle layer.

Fig.4 Relationship between number of effective hidden units and number of hidden units

4.6 Network precision

Fig.7 shows the comparison between kiss point position calculated by this network model and that by the inversed algorithm using measured rolling force. It can be observed from Fig.7 that the network model exhibits high precision.

Fig.5 Relationship between square root of mean square error of prediction (SEP) and M

Fig.6 Relationship between square root of mean square error of prediction (SEP) and M

Fig.7 Comparison between kiss point position calculated by model and inversely calculated by measured rolling force

5 Verification of casting rolling force model

The experimental material was 1Cr17. The model parameters are listed in Table 1. From Fig.8, it can be seen that calculation precision of test data of 94.67% is less than ±7.0%, which illustrates that the rolling force model has high precision.

Table 1 Model parameters

Fig.8 Comparison between calculated and measured rolling forces

6 Conclusions

(1) A model for rolling force prediction is built in which solution zone is divided into two parts by kiss point position during casting strip: liquid zone and mushy zone. Navier-Stokes equation is introduced as the fundamental equation to discuss the rolling force, and the velocity filed is established through stream function. Solid phase fraction, viscosity and temperature distribution model is used, and Bayesian regularization is introduced into the training of feedforward neural networks in calculation of kiss point position.

(2) During rolling force prediction for casting strip of 1Cr17 ferritic stainless steel using this model, calculation precision of test data of 94.67% is less than ±7.0%. This model has high precision for prediction.

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Foundation item: Project(2004CB619108) supported by National Basic Research Program of China

Received date: 2010-04-01; Accepted date: 2010-04-01

Corresponding author: CAO Guang-ming, PhD; Tel: +86-13504187137; E-mail: caogm@ral.neu.edu.cn

(Edited by LIU Hua-sen)