Influence of Superheat on Macrosegregation in Continuously Cast Steel Billet from Statistical Maximum Viewpoint

Abstract

The statistics of extreme values (SEV) and the generalized Pareto distribution (GPD) are introduced to obtain the predicted maximum carbon content and represent the macrosegregation degree along the casting direction in continuous casting billets (represented by 82B cord steel). Subsequently, the influence mechanism of the superheat on macrosegregation has been investigated. It’s found that the SEV and the GPD methods can correctly obtain the predicted maximum carbon content and the results obtained by the SEV method increase linearly with the return period, while the GPD method can calculate the upper limit value of carbon content. The stochastic mathematical methods can get rid of the limitations of the sampling length and analysis area. During the exploration of the superheat influence mechanism based on the results obtained by these two methods, lower superheat can increase the secondary dendritic arm spacing (SDAS) and make the enriched carbon element easier concentrating on the centerline, resulting in centerline segregation and the increase of the corresponding result in the centerline. Under higher superheat condition, the segregation will be mainly represented as V-shaped segregation due to the decrease of the SDAS, leading to the increase of corresponding result in the off-center position.

1. Introduction

High carbon steel billet is an important raw material for high-strength cord products. However, carbon macrosegregation is prone to occur during the continuous casting process due to the high carbon content and low partition coefficient,¹⁾ which seriously affect the quality uniformity of high-strength cord products. During the final stage of solidification, the flow of enriched solute will lead to macrosegregation defects in the equiaxed crystal zone²⁾ and these defects will be presented as V-shaped segregation and centerline segregation along the casting direction, which cannot be detected in the transverse section easily and always indicate the fluctuation of carbon content in the vertical section. Meanwhile, performance failures of product can always be caused by this kind of content fluctuation, especially by the maximum value in the content distribution. Thus, the research on the macrosegregation defects in the vertical section of high carbon steel billets is important to improve the cord product quality.

During the continuous casting process, the superheat can affect the formation of macrosegregation in billets by changing the solidification process and the solidification structure,^3,4) yet there is a dispute about the influence mechanism of superheat. Previous researches showed that lower superheat can increase the equiaxed crystal ratio and reduce the segregation degree.^1,5) With the development to depth, some scholars found that a higher superheat can improve the central quality and reduce the central segregation degree of billets.^6,7) The above researches on the influence of superheat are usually simple analysis on certain locations. In the research process, different sampling positions and sample sizes will result in differences in evaluation results. Therefore, systematic researches on the influence mechanism of superheat in the continuous casting process are still necessary. Meanwhile, previous researches on the macrosegregation in the vertical section are usually observations or qualitative comparisons,^8,9,10) lacking in systematic and quantitative calculations. Hou^11,12) investigated the periodicity of carbon distribution in high carbon steel billets and the ARMA (Auto Regressive Moving Average) model¹³⁾ was introduced to analyze the fluctuation characteristics of carbon element content along the casting direction. However, the quantitative prediction of the maximum value of carbon element content along the casting direction is still unclear, which can represent the most serious segregation degree and affect the quality uniformity of cord products. On the other hand, the drilling hole sampling and chemical analysis are the most used methods in the actual production process to judge the quality of billets and segregation degree. Nevertheless, the judgment may be fluctuant and non-repeatable because of the limitations of the sampling length and analysis area. Thus, appropriate methods that can effectively evaluate the macrosegregation degree based on limited samples and analysis areas are required. On this basis, research on the influence mechanism of superheat on the macrosegregation defects is of great significance to control the quality uniformity of the billets and improve the quality of cord products.

Based on above background, this paper tried to introduce the stochastic mathematical methods including the statistics of extreme values (SEV) and generalized Pareto distribution (GPD) to obtain the predicted maximum value of carbon element content (abbreviated as C_PM) in the equiaxed crystal zone along the casting direction of high carbon steel billets and represent the segregation degree. The SEV method and GPD method are classic statistical methods at present, which have been applied in many fields of materials research.¹⁴⁾ Murakami^15,16) introduced the SEV method in the estimation of the maximum size of inclusions in metal and the results were used to control the quality of materials and predict a scatter band of fatigue strength. Scarf¹⁷⁾ investigated the application of the extreme value theory into corrosion engineering to predict the maximum pit depth in a certain area, which often fails materials. Sun¹⁸⁾ applied the SEV method and the GPD method to estimate the maximum size of inclusions in ultra-clean GCr15 steel, on this basis, the relationship between the maximum size of inclusions and bearing fatigue life has been explored. However, the application of this kind of stochastic mathematical methods to calculate the C_PM in the equiaxed crystal zone of billets has not been reported. Therefore, in this paper, the C_PM in equiaxed crystal zones will be calculated and discussed by introducing different stochastic mathematical methods. Subsequently, the influence mechanism of superheat on the segregation defects will be investigated based on the calculated results and actual solidification structure characteristics.

2. Materials and Methods

2.1. Materials

82B cord steel (steel grade: SWRH82B) with a size of 150 mm×150 mm was used as the research object. 82B cord steel billets are usually used to produce steel wire products through cold drawing. However, the existence of carbon macrosegregation will affect the uniformity of central phase structure of wire rod and increase the risk of fracture failure.¹⁹⁾ Thus, the research on the macrosegregation in 82B cord steel billets is significant. The main chemical composition (mass%) of 82B cord steel billets is C0.82, Si0.22, Mn0.50, P≤0.008, S≤0.008, and Fe balance. Two billet samples were obtained in the actual production field under different process parameters as shown in Table 1. In actual production process, temperature of molten steel at a same position in the tundish was measured in order to obtain accurate casting superheat. The only process difference between of two billets is the superheat. One is 34°C, and the other is 29°C. Methods of drilling hole sampling and chemical analysis were applied to obtain the original carbon distribution in the equiaxed crystal zone of billet Nos. 1 and 2. The sampling position is shown as Fig. 1, in which line AB is the centerline of the equiaxed crystal zone in the vertical section (abbreviated as centerline) and line CD represent the sideward position 20 mm from the centerline in the equiaxed crystal zone (abbreviated as off-center position). During the drilling process, the drilling diameter is 5 mm with a depth of 10 mm and the number of drilled samples is 30 in each position. After drilling hole sampling, the carbon distribution was obtained by chemical analysis.

Table 1. Main process parameters of 82B cord steel.

Sample	Superheat/°C	Casting speed/ (m min−1)	Cooling intensity in third section of secondary cooling/(m3 h−1)
No. 1	34	1.9	1.7
No. 2	29	1.9	1.7

Fig. 1.

Drilling hole sampling positions. (Online version in color.)

2.2. Methods

2.2.1. Statistics of Extreme Values (SEV)

The statistics of extreme values (SEV) is based on Gumbel distribution²⁰⁾ and the Gumbel distribution function G(x) is as follows:   

$$G\left(x\right)=exp\left\{\text{-exp}\left[-\left(x-\lambda \right)/\alpha \right]\right\}$$(1)

α and λ are the undetermined parameters.²¹⁾ In this paper, the carbon content x_i in the N samples is measured and the cumulative probability of the carbon content x_i is as Eq. (2). To calculate the C_PM in the billet with a length of L, parameter T was defined as Eq. (3), in which L₀ is diameter of individual sample point during drilling hole sampling. The C_PM in the billet can be represented as the Eqs. (4) and (5), where x_V=C_PM; $\hat{\alpha }$ and $\hat{\lambda }$ are the estimated value of α and λ, respectively. The value of the parameters α and λ could be obtained by the least-squares method or maximum likelihood estimation method.²²⁾ According to Murakami’s research,²³⁾ the likelihood function estimation is more effective and the results are more accurate. Therefore, in this paper, the least-squares method will be used first to obtain the original value of α and λ and after that the likelihood function estimation will be used to calculate the final value of $\hat{\alpha }$ and $\hat{\lambda }$. The likelihood function expression is shown as Eq. (6):   

$$G\left(x_{\text{i}}\right)=i/\left(N+1\right)$$(2)

  

$$T=L/L_0$$(3)

  

$$x_{\text{V}}=\hat{\alpha }y+\hat{\lambda }$$(4)

  

$$y=-ln\left\{-ln\left[G\left(x_{\text{V}}\right)\right]\right\}=-ln\left\{-ln\left[1-1/T\right]\right\}$$(5)

  

$$L\left(x_1,x_2\mathrm{...}{x}_{\text{N}},\lambda ,\alpha \right)=\prod _{i=1}^N\frac{1}{\alpha }exp\left\{-\left[\frac{x_{\text{i}}-\lambda }{\alpha }+exp\left(-\frac{x_{\text{i}}-\lambda }{\alpha }\right)\right]\right\}$$(6)

2.2.2. Generalized Pareto Distribution (GPD)

The GPD method is used to analyze random data above a certain threshold value, which has been applied to calculate the maximum size of inclusions in clean steel.²⁴⁾ In this paper, assuming that the carbon distribution has the threshold value u and the distribution function of the carbon content larger than the threshold value can be given by the generalized Pareto distribution. The generalized Pareto distribution function F(x) is shown as Eq. (7), in which σ is size parameter and ξ is shape parameter. The estimated values of σ and ξ can be obtained by the likelihood function and the density equation of the corresponding likelihood function is as Eq. (8), in which x_i is the carbon content larger than the threshold value u.   

$$F\left(x\right)=1-{\left(1+\xi \left(x-u\right)/\sigma \right)}^{-1/\xi }$$(7)

  

$$L=\stackrel{k}{\prod _{i=1}}\frac{1}{\sigma }{\left\{1+\frac{\xi \left(x_{\text{i}}-u\right)}{\sigma }\right\}}^{-\left(1/\xi \right)-1}$$(8)

In the calculation process of the C_PM in the billets with length of L. Assuming that N_L(u) is the expected value of carbon content larger than the threshold value at unit length and x_L is the maximum value of carbon element content in the billet with length of L, then the relationship can be given as Eq. (9), where $\hat{\sigma }$ and $\hat{\xi }$ represent the estimated values of σ and ξ. Combined with Eq. (7), the C_PM can be represented as the Eq. (10). When L is large enough and $\hat{\xi }$ is negative, the equation can be approximated as Eq. (11) and x_L′ is the C_PM calculated by the GPD method.   

$$N_{\text{L}}\left(u\right)L\cdot \left(1-F\left(x_{\text{L}}\right)\right)=1$$(9)

  

$$x_{\text{L}}=u-\frac{\hat{\sigma }}{\hat{\xi }}\left\{1-{\left(N_{\text{L}}\left(u\right)L\right)}^{\hat{\xi }}\right\}$$(10)

  

$$x_{\text{L}}\prime =u-\frac{\hat{\sigma }}{\hat{\xi }}$$(11)

3. Results and Discussion

3.1. Original Carbon Distribution

Carbon distribution of different positions in the equiaxed crystal zone obtained by drilling hole sampling and chemical analysis is shown in Fig. 2. It can be seen that the carbon distribution along the casting direction have strong fluctuation and the difference between adjacent samples may be large. In this case, it will be inaccurate if the actual maximum or minimum value of carbon content is selected as the standard for judging the segregation degree of billets and the actual maximum or minimum value of carbon content can not represent the quality of billets due to the limitations of sampling length and analysis area. In the continuous casting process, the macroscopic solidification conditions in the equiaxed crystal zone of the billets can be approximately the same. In this case, the samples obtained by drilling hole sampling can be regarded as solidification units formed along the casting direction under the same conditions. Therefore, the SEV method and the GPD method can be used to calculate the C_PM based on the carbon content in the samples and represent the macrosegregation degree in the equiaxed crystal zone of billets, which can provide guidance for the quality judgment of steel wire products.

Fig. 2.

Carbon distributions along casting direction in the equiaxed crystal zone of billets Nos. 1 and 2. (Online version in color.)

3.2. Statistical Maximum Value of Carbon Element Content

The C_PM in the centerline and off-center position of 82B cord steel billets were obtained by the SEV method at first. The values of parameters were determined by the likelihood estimation method and the results are presented in Table 2. Depending on Eqs. (4) and (5), the C_PM along the casting direction can be obtained. The cumulative probability G(x_V) in Eq. (5) represents the return level of the C_PM, namely the probability that the actual carbon element content is not more than the calculated C_PM and it is also a representation of the calculation length of billets. For instance, when the cumulative probability G(x_V) is 99.9%, the calculation according to Eqs. (3) and (5) shows that the return period T =1000. In this case, the calculated x_V represents the C_PM in the billet with a length of 1000_L0 and the actual carbon element content will all less than x_V. Similarly, when the cumulative probability G(x_V) is 99.99%, the calculated result represents the C_PM in the billet with a length of 10000_L0.

Table 2. Calculated parameters of the SEV method in different positions.

Sample	Location	$\hat{\lambda }$	$\hat{\alpha }$
No. 1	Center line	0.858	0.037
	Off center position	0.827	0.016
No. 2	Center line	0.829	0.041
	Off center position	0.825	0.013

The C_PM under different cumulative probabilities obtained by the SEV method are shown in Fig. 3 and the results indicate that there is a linear relationship between the maximum value calculated by the SEV method and the return period index, namely the C_PM correspond to the length of billet and will linearly increase with the billet length index increase.

Fig. 3.

C_PM under different cumulative probabilities based on the SEV method. (Online version in color.)

The GPD method can calculate the C_PM by analyzing the data greater than the threshold value and the determination of the threshold value is an important aspect of the GPD method. The carbon element content in two billet samples was analyzed by the GPD method to obtain the values of threshold u, estimated values of shape parameter ξ and size parameter σ as presented in Table 3. Then, the C_PM along the casting direction in billets can be calculated according to Eq. (11).

Table 3. Calculated parameters of the GPD method in different positions.

Sample	Location	u	$\hat{\xi }$	$\hat{\sigma }$
No. 1	Center line	0.800	−0.660	0.128
	Off center position	0.824	−0.651	0.036
No. 2	Center line	0.764	−0.598	0.141
	Off center position	0.814	−0.742	0.037

In the calculation process of the GPD method, there is also a return level related to the length of billets. Assuming the return level is cumulative probability p, the predicted maximum value of the carbon element content under cumulative probability p can be expressed as Eq. (12), in which the N is the number of the drilling samples and N_u is the number of times the carbon content in the samples above the threshold value. The probability will be p when the actual carbon element content around the centerline or off-center position are all less than C_PM(p). The C_PM calculated by the GPD method under different cumulative probabilities are illustrated in Fig. 4. It can be seen that the C_PM under different cumulative probabilities increases gradually and approach to the value obtained by Eq. (11), that is, the C_PM calculated by the GPD method is the upper limit value of carbon element content when the cumulative probability approach to 100%.   

$$C_{\text{PM}(p)}=u+\frac{\hat{\sigma }}{\hat{\xi }}\left\{{\left[\frac{N}{N_u}\left(1-p\right)\right]}^{-\hat{\xi }}-1\right\}$$(12)

Fig. 4.

C_PM under different cumulative probabilities based on the GPD method. (Online version in color.)

It can be found in Figs. 3(a) and 4(a) that the changing trend of the C_PM in the centerline of billet Nos. 1 and 2 is crossed during the calculation process, that is, with the increase of the return period, the C_PM in the centerline of the billet No. 2 changed more intensively and finally exceed that of the billet No. 1. The C_PM calculated by the SEV method and the GPD method are not only related to the actual carbon element content in billets, but also the carbon discrete degree. To explain the relationship between the C_PM and the carbon discrete degree, the carbon element content in the centerline of billets Nos. 1 and 2 was analyzed. Figures 5 and 6 are the calculation of the original parameters of the SEV method and the GPD fitting of the carbon element content above the threshold value, respectively. It can be observed that both the determination of parameters in the SEV method and the GPD fitting are all related to the carbon distribution, especially the fluctuation between the actual maximum and minimum values of the carbon element content. The larger the discrete degree of the carbon distribution in the actual billet, the stronger the volatility of the carbon element content. In the long return period, the strongly fluctuant carbon distribution is more possible to lead a larger C_PM. Figure 7 and Table 4 are the carbon distribution interval and carbon distribution characteristics in the centerline of billets Nos. 1 and 2, respectively. It can be seen that the carbon distribution range in the centerline of billet No. 2 is larger and more discrete compared with that in billet No. 1, which makes it more possible to obtain a larger C_PM in centerline by these two methods. On the other hand, although the changing trend of the C_PM in the centerline of the two billet samples in Figs. 3(a) and 4(a) is crossed, the cross position is different. In the GPD method, the crossing point of carbon maximum value in centerline appears earlier and the upper limit value is more obvious, which means that the GPD method will reflect the change of the C_PM more timely. As for the SEV method, the crossing point in the centerline appears later and only crossed when the return period is large enough, that is, the SEV method reflects the discrete degree of the actual carbon distribution and the change of the C_PM more slowly, which may lead to errors in the calculation process.

Fig. 5.

Original undetermined parameters in SEV method of billets Nos. 1(a) and 2(b) obtained by plotting. (Online version in color.)

Fig. 6.

Fitting of carbon element billets Nos. 1(a) and 2(b) content exceeding threshold value by GPD method. (Online version in color.)

Fig. 7.

Carbon distribution in different intervals of billets Nos. 1 and 2. (Online version in color.)

Table 4. Distribution characteristics of carbon element content in the centerline of billets Nos. 1 and 2.

Sample	Range/%	Mean-square deviation
No. 1	0.188	0.043
No. 2	0.221	0.049

Based on the analysis of the results calculated by these two methods, it can be seen that the SEV method and the GPD method can all obtain the C_PM in the equiaxed crystal zone of continuous casting billets and the changing trend of the C_PM in different cumulative probabilities are related to the discrete degree of the actual carbon element distribution in the billets. By comparing the changing trend of C_PM, it is found that there is no specific upper limit value for the C_PM under different cumulative probabilities calculated by the SEV method and the C_PM increase linearly with the increase of the return period index. As for the GPD method, the calculated C_PM will approach a specific upper limit value and this is consistent with the actual continuous casting process. According to analyzing the crossing point of the C_PM in the centerline of billets Nos. 1 and 2, it can be seen that the crossing point is caused by the different discrete degrees of carbon distribution in the centerline of billets Nos. 1 and 2. Meanwhile, the GPD method can reflect the discrete degree of carbon distribution and changing trend of the C_PM more timely and the distinction of the maximum value is more obvious, while the SEV method will be relatively slow to reflect the change of the C_PM and it is only in a longer return period that the crossing point is reflected.

3.3. Influence Mechanism of Superheat on Segregation

Superheat in the continuous casting process is a major factor affecting the morphology of the solidification structure and the formation of macrosegregation. Thus, it is necessary to make further explorations to find out the influence mechanism of superheat on the segregation defects and the C_PM along the casting direction in billets. As mentioned above, the C_PM obtained by the GPD method is the upper limit value of carbon element content when the cumulative probability is close to 100%. On the contrary, the results calculated by the SEV method increase linearly with the increase of the return period index and the changing trend of C_PM is reflected when the return period is large enough. It can be seen from Fig. 3(a) that when the cumulative probability is greater than 99.99%, the changing trend of the C_PM in the centerline obtained by the SEV method will not change. Therefore, to compare these two methods and reduce the comparison error, this paper will calculate the C_PM under the SEV cumulative probability of 99.99% and compare the result with the upper limit value of carbon content obtained by the GPD method. On this basis, the influence mechanism of superheat on the segregation defects and carbon distribution will be explored. Figures 8 and 9 show the changing trend of C_PM with superheat in the centerline and off-center position in billets Nos. 1 and 2. It can be seen that the changing trend with the decrease of superheat is the same. That is, the C_PM increase in centerline with the decrease of superheat and decrease in the off-center position of the equiaxed crystal zone.

Fig. 8.

Relationship between superheat and the C_PM calculated by SEV method in the center line (a) and off center position (b) of billets Nos. 1 and 2. (Online version in color.)

Fig. 9.

Relationship between superheat and the C_PM calculated by GPD method in the center line (a) and off center position (b) of billets Nos. 1 and 2. (Online version in color.)

To investigate the influence mechanism of superheat on macrosegregation defects in equiaxed crystal zone, it’s necessary to analyze the results of C_PM and the solidification structure characteristics of continuous casting billets. The solidification structure characteristics of the longitudinal section can not be obtained effectively because of the drilling hole sampling. Therefore, the actual solidification structure in transverse section obtained by hot pickling experiment has been used to characterize the solidification characteristics of billets Nos. 1 and 2 under different superheat conditions. At the same time, the equiaxed crystal ratio and secondary dendritic arm spacing (SDAS) of the two billet samples were measured and calculated. The actual solidification structure in the cross-section of billets Nos. 1 and 2 are shown in Fig. 10. It can be seen that under high superheat condition, the segregation degree in the transverse section of the billet is less and the segregation points are more dispersed (point segregation in the equiaxed crystal zone will be shown as V-shaped segregation in vertical section). Under the low superheat condition, the segregation degree in equiaxed crystal zone is more serious and the segregation point is concentrated in the central region (the concentrated segregation defects are usually shown as centerline segregation in vertical section). The calculated results of the solidification structure characteristics of billets Nos. 1 and 2 are shown in Table 5. In the calculation process, 30 regions with a size of 10 mm×10 mm have been taken in the equiaxed crystal zone to calculate the SDAS. In each region, the linear intercept method was used to measure 3–4 typical dendrites. The average value of the SDAS in 30 regions was obtained and used to represent the average SDAS in the equiased crystal zone. According to Table 5, under the condition of low superheat, the equiaxed crystal ratio and the SDAS are larger than that under high superheat condition, which has the same trend with the calculation results in previous researches.^8,25)

Fig. 10.

The comparison of solidification structure in cross section of billets under high superheat (a) and low superheat (b). (Online version in color.)

Table 5. Central equiaxed crystal ratio and the SDAS of billets Nos. 1 and 2.

Sample	Superheat/°C	Equiaxed crystal ratio/%	SDAS/μm
No. 1	34	19.34	224.31
No. 2	29	23.96	230.42

During the continuous casting process, there is a relationship between the SDAS and the central solidification time (the time required to reduce the temperature in the centerline of billet from the liquidus temperature to the solidus temperature).²⁶⁾ Under low superheat condition, the molten steel in the mold will reach the liquidus temperature more quickly, resulting in the formation of a thicker billet shell. The thermal resistance in the later stage of solidification process in continuous casting mainly comes from the formed shell and not sensitive to the superheat changing.^6,27) Thicker billet shell will increase the central solidification time and decrease the cooling rate, leading to an increase of SDAS according to the Eq. (13), in which d₂ represent the SDAS and Gv is cooling rate. The SDAS is an important factor affecting the liquid flow and the formation of segregation defects in the final stage of the continuous casting process. Higher superheat will lead to smaller SDAS and lower permeability in equiaxed crystal zone, which can hinder the flow of solute-enriched liquid phase from off-center position to the centerline and the V-shaped segregation in the equiaxed crystal zone will be aggravated. Thus, the C_PM in the off-center position is greater than that under low superheat. On the contrary, larger SDAS under lower superheat increased the concentration of the solute-enriched liquid phase to the centerline and aggravate the centerline segregation, leading to a greater C_PM in centerline under low superheat. These results are consistent with previous researches^27,28) and also correspond to the actual solidification structure in Fig. 10. At the same time, the correspondence between the results and the solidification structure can also indicate the accuracy of the C_PM obtained by the SEV method and the GPD method.   

$$d_2=\beta {\left(Gv\right)}^{-1/3}$$(13)

Based on the above analysis, the influence mechanism of superheat on the formation of solidification structure and segregation defects in the billets can be summarized as Fig. 11, where T_L represents the liquidus temperature of molten steel, T_S represents the solidus temperature, t_LS1 and t_LS2 represent the central solidification time under higher superheat and lower superheat, respectively. With the decrease of superheat in the continuous casting process, the equiaxed crystal zone in the billet is enlarged and the equiaxed crystal ratio is increased. However, lower superheat can prolong the central solidification time (t_LS2) and increase the SDAS, leading to the concentration of the solute-enriched liquid phase in the centerline and aggravate the centerline segregation. Under a higher superheat, the reduction of central solidification time (t_LS1) will decrease the SDAS and increase the flow resistance of the solute-enriched liquid phase, which can aggravate the formation of V-shaped segregation in equiaxed crystal zone. Thus, the C_PM in the off-center position is greater than that under low superheat.

Fig. 11.

The influence mechanism of superheat on the segregation in equiaxed crystal zone under high superheat (a) and low superheat (b) (“⊕” represent the flow of liquid phase in dendrites). (Online version in color.)

In summary, both the SEV method and the GPD method can quantitatively calculate the C_PM in high carbon steel continuous casting billets and the results are related to the actual carbon distribution in billets. The calculation of the C_PM can quantitatively reflect the macrosegregation degree of the 82B cord steel billets and provide the basis for improving the quality of the billets and the follow-up steel wire products. The C_PM calculated by the SEV method does not have an upper limit value and the calculated results increase linearly with the increase of the return period index. As for the GPD method, the C_PM is a specific upper limit value and this is consistent with the actual continuous casting process. By comparing the calculation results under different cumulative probabilities, it is found that the GPD method can reflect the changing trend of C_PM in a more timely manner and the SEV method reflects the actual carbon dispersion degree and the changing trend of C_PM slowly, which may lead to errors in the calculation process. During the continuous casting process of high carbon steel billets, the equiaxed crystal zone is enlarged and the equiaxed crystal ratio is increased with the decrease of superheat. Nevertheless, the SDAS in equiaxed crystal region increases under low superheat and enriched carbon element flow to the centerline of the billets, which aggravates the formation of the centerline segregation. The existence of centerline segregation can increase the fluctuation degree of carbon element content and the calculated C_PM in the centerline. Under high superheat conditions, the decrease of SDAS in the equiaxed crystal zone will increase the resistance of the liquid flow, which can aggravate the formation of V-shaped segregation. Therefore, the C_PM at the off-center position of the equiaxed crystal zone is greater.

4. Conclusion

(1) Both the SEV method and the GPD method can quantitatively obtain the predicted maximum value of carbon element content (C_PM) in high carbon steel continuous casting billets and the results are related to the actual carbon discrete degree in billets. The result calculated by the SEV method increases linearly with the increase of the return period index, while the result calculated by the GPD method is a definite upper limit value. According to the analysis of the C_PM under different cumulative probabilities, it is found that compared with the SEV method, the GPD method can reflect the changing trend of the C_PM timely, which is beneficial to reduce the calculation errors.

(2) By comparing the C_PM and the solidification structure characteristics, it can be seen that the decrease of the superheat will increase the equiaxed crystal ratio of the billets, but at the same time, the increase of the central solidification time of the billets will increase the secondary dendrite arm spacing (SDAS) and reduce the resistance of the solute-enriched liquid phase to the center. The enrichment of carbon element in the centerline will result in centerline segregation and increase of the corresponding centerline C_PM. Notwithstanding, under high superheat condition, the SDAS in the equiaxed crystal zone decreases and the liquid flow resistance increases, which reduce the formation of centerline segregation. In this case, the segregation defect in the vertical section will be mainly represented as V-shaped segregation and the off-center C_PM has a higher value than that under low superheat.

(3) In the actual production process of high carbon steel billets, the C_PM in the billets can be calculated to evaluate the segregation degree based on the stochastic mathematical methods (SEV, GPD, etc.) and on-site chemical analysis samples, which can greatly improve the use of samples. This paper can provide a new method for the calculation of the maximum content of segregation elements in billets and the control of billet quality and segregation.

Acknowledgements

The authors are very grateful for support from United Funds between National Natural Science Foundation and Baowu Steel Group Corporation Limited from China (No.U1860101) and Chongqing Fundamental Research and Cutting-Edge Technology Funds (No.cstc2017jcyjAX0019).

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