Mathematical statistics for measurement of steel temperature in steel-pouring ladle and tundish at steel continuous casting

The article considers the temperature distribution in steel during its continuous casting. Temperatures were measured sequentially in the steelpouring ladle (one measurement) and in the tundish (two measurements) using a platinum-platinum-rhodium thermocouple with an accuracy of ±4 °C. We have analyzed the results of 170 casts of two steel grades: 5SP and 35GS. The type of temperatures set distribution was verified on the basis of three goodness-of-fit criteria: Pearson’s χ-square criterion, λ Kolmogorov-Smirnov criterion and W Shapiro-Wilk criterion. The results obtained are consistent with the physical picture of steel casting. The metal in steel-pouring ladle is practically in a stable state and is subject only to natural cooling through the lining, top and ladle body. In the variant of analyzing a sample of temperature values in tundish at the first and second measurements, the hypothesis of normal distribution should be rejected. Here, the steel temperature depends on a number of parameters, including the feed rate and casting rate, feed time and composition of slag-forming and heat-insulating mixtures, etc. Attempts to establish the relationship between the steel temperatures of in steel-pouring ladle and tundish were unsuccessful. Considering the temperature measurement in tundish as two sequential data arrays, the first of which is an argument, and the second is a function, a linear relationship between these arrays was established. This relationship between the first and second temperature measurements in the tundish can be used to estimate the steel final temperature at thermocouple readout, including in the event of a failure. The results of the work can be used in development of a mathematical model of steel casting.

1. Amlin A.V., Shchipanov S.S., Amelin Al.V., Foigt D.B. Development of steel continuous casting in JSC “EVRAZ-ZSMK”. Stal’. 2019, no. 7, pp. 14–16. (In Russ.).

2. Han Z., Li Y., Yang M., Yuan Q., Ba L., Xu E. Digital twin-driven 3D visualization monitoring and traceability system for general parts in continuous casting machine. Journal of Advanced Mechanical Design, Systems, and Manufacturing. 2020, vol. 14, no. 7, article 0100. https://doi.org/10.1299/jamdsm.2020jamdsm0100

3. Jong-Kyu Yoon. Applications of numerical simulation to continuous casting technology. ISIJ International. 2008, vol. 48, no. 7, pp. 879–884. https://doi.org/10.2355/isijinternational.48.879

4. Jiaocheng Ma, Zhi Xie, Guanglin Jia. Applying of real-time heat transfer and solidification model on the dynamic control system of billet continuous casting. ISIJ International. 2008, vol. 48, no. 12, pp. 1722–1727. https://doi.org/10.2355/isijinternational.48.1722

5. Grip Carl-Erik. Simple model for prediction of temperatures in an L-shaped tundish – Verification by continuous temperature measurements. ISIJ International. 1998, vol. 38, no. 7, pp. 704–713. https://doi.org/10.2355/isijinternational.38.704

6. Fan C.-M., Hwang W.-S. Mathematical modeling of fluid flow phenomena during tundish filling and subsequent initial casting operation in steel continuous casting process. ISIJ International. 2000, vol. 40, no. 11, pp. 1105-1114. https://doi.org/10.2355/isijinternational.40.1105

7. Pardeshi R., Basak S., Singh A.K., Basu B., Mahashabde V., Roe S.K., Kumar S. Mathematical modeling of the tundish of a single-strand slab caster. ISIJ International. 2004, vol. 44, no. 9, pp. 1534–1540. https://doi.org/10.2355/isijinternational.44.1534

8. Liu S.-X., Yang X.-M., Du L., Li L., Liu C.-Z. Hydrodynamic and mathematical simulations of flow field and temperature profile in an asymmetrical t-type single-strand continuous casting tundish. ISIJ International. 2008, vol. 48, no. 12, pp. 1712–1721. https://doi.org/10.2355/isijinternational.48.1712

9. El’darkhanov A.S., Nuradinov A.S., Baranova V.N. Some aspects of external influences application during continuous casting of steel. Stal’. 2015, no. 10, pp. 17–20. (In Russ.).

10. Botnikov S.A., Khlybov O.S., Kostychev A.N. Development of a steel temperature prediction model in a steel ladle and tundish in a casting and rolling complex. Steel in Translation. 2019, vol. 48, no. 10, pp. 688–694. https://doi.org/10.3103/S096709121910005X

11. Timoshpol’skii V.I., Trusova I.A. The technological improvement for continuous casting of section steel, and the measuring methods for the temperature during solidification and cooling, Report I. Steel in Translation. 2019, vol. 49, no. 11, pp. 783–788. https://doi.org/10.3103/S0967091219110147

12. Van Ende M.-A., Jung I.-H. Development of a thermodynamic database for mold flux and application to the continuous casting process. ISIJ International. 2014, vol. 54, no. 3, pp. 489–495. https://doi.org/10.2355/isijinternational.54.489

13. Klimanchuk V.V., Bochek A.P., Lavrinishin S.A., etc. Efficiency of protection of continuously casted metal. Stal’. 2007, no. 1, pp. 20-22. (In Russ.).

14. Toptygin A.M., Polozov E.G., Aizin Yu.M., Neklyudov I.V. Improving protective slag-forming mixtures for intermediate ladles of continuous-casting machines. Steel in Translation. 2007, vol. 37, no. 3, pp. 266–270. https://doi.org/10.3103/S0967091207030254

15. Kapitanov V.A., Kuklev A.V., Polozov E.G. Heat-insulating properties of slag mixtures for the intermediate ladle. Steel in Translation. 2009, vol. 39, no. 1, pp. 50–52. https://doi.org/10.3103/S0967091209010124

16. Vil’danov S.K., Likhodievskii A.V., Pyrikov A.N. Development and introduction of energy saving materials for steel pouring. Refractories and Industrial Ceramics. 2011, vol. 52, no. 4, pp. 237–239. https://doi.org/10.1007/s11148-011-9404-z

17. Vil’danov S.K. Development and implementation of heat-insulating and slag-forming materials of Isotherm-1600 series. Stal’. 2018, no. 9, pp. 17–22. (In Russ.).

18. Vil’danov S.K., Likhodievskii A.V. Heat-insulating and protective mixture for bath level in CCM tundish. Patent RF no. 2334587. Bulleten’ izobretenii. 2008, no. 27. (In Russ.).

19. il’danov S.K., Likhodievskii A.V., Pyrikov A.N. Heat-insulating thermo-expanding mixture. Patent RF no. 2464122. Bulleten’ izobretenii. 2012, no. 29. (In Russ.).

20. Palii I.A. Applied Statistics. Moscow: Vysshaya shkola, 2004, 175 p.

21. Pearson E.S., Hartley H.O. Biometrika Tables for Statisticians. Cambridge University Press, 1954, vol. 1.

22. Ivchenko G.I., Medvedev Yu.I. Mathematical Statistics. Moscow: Vysshaya shkola, 1984, 247 p. (In Russ.).

23. Cox D.R., Hinckley D.V. Theoretical Statistics. Chapman and Hall/ CRC, 1979, 528 p.

24. Lehman E.L. Testing Statistical Hypotheses. New York: John Wiley and Sons Inc., London: Chapman and Hall, 1959, 361 p.

25. Kateman G., Pijpers F.W. Quality Control in Analytical Chemistry. Wiley, 1981, 276 p.

26. Karlin S. First Course in Stochastic Processes. New York: Academic Press, 1966, 536 p.

27. Hald A. Statistical Theory with Engineering Applications. New York: John Wiley, 1952.

28. Brownlee K.A. Statistical Theory and Methodology in Science and Engineering. New York: John Wiley, 1965.

29. Kolmogorov A.N. Probability Theory and Mathematical Statistics. Moscow: Nauka, 1986, 535 p. (In Russ.).

30. Johnson Norman L., Leone Fred C. Statistics and Experimental Design in Engineering and the Physical Sciences. New York, etc.: John Wiley, 1977.

31. Bocharov P.P., Pechinkin A.V. Mathematical Statistics. Moscow: Rossiiskii universitet druzhby narodov, 1994, 164 p. (In Russ.).

32. Shapiro S.S., Wilk M.B. An analysis of variance test for normality (complete samples). Biometrika. 1965, vol. 52, pp. 591–611. https://doi.org/10.2307/2333709

33. Cochran W.G. The χ2 test of goodness of fit. The Annals of Mathematical Statistics. 1952, vol. 23, pp. 315–345.

34. Williams C.A. On the choice of the number and width of classes for the Chi-Square test goodness of fit. Journal of the American Statistical Association. 1950, vol. 45, pp. 77–86.

35. Hahn G.J., Shapiro S.S. Statistical Models in Engineering. Wiley, New York, 1967, 395 p.