# High Quality Steel Casting by Using Advanced Mathematical Methods

^{*}

## Abstract

**:**

## 1. Introduction

^{®}(BrDSM) and advanced optimal control algorithm based on the fuzzy logic (FL-BrDSM). This unique combination represents a tool for achievement of high steel quality products. Besides of the numerical and fuzzy regulation models, special attention must be concentrated to proper setting of thermophysical parameters of investigated steel grade, experimental measurement of boundary conditions and statistical evaluations of the real casting data.

## 2. Solidification Model—BrDSM

_{eff}is the effective thermal conductivity (W/mK); T is the temperature (K); T

_{pouring}is the pouring temperature (K); h is the specific enthalpy (J/kg); ρ is the density (kg/m

^{3}); τ is the time (s); v

_{cast}is the casting speed (m/min); z is the direction of casting (m), T

_{rol}is the roller temperature; T

_{water}is the cooling water temperature; T

_{amb}is the ambient temperature; $\dot{m}$

_{water}is the mass water flow in the mould (kg/s); c

_{water}is the specific heat capacity of water (J/kgK); htc is the heat transfer coefficient beneath spraying surface (W/m

^{2}K); σ is the Stefan-Boltzman constant (W/m

^{2}K

^{4}) and ε is the emissivity of the slab surface (-).

_{s}is the solid fraction (-). The Enthalpy method was used for modelling the solidification process, see Mauder et al. [10]. The Enthalpy method is robust method because it ensures energy conservation and there is no discontinuity at either the liquidus or the solidus temperatures because the solidification/melting path is characterized strictly by decreasing/increasing enthalpy. In the Equation (1) the enthalpy is calculated in the first step as the primary variable and the temperature is calculated from a defined enthalpy-temperature relationship in the second step Equation (8).

_{eff}. The thermal conductivity is increased by the flow of liquid steel at different distances from the meniscus, where k

_{eff}is represented by Zhang et al. [12] as follows:

_{sol}is the temperature of solidus (K); T

_{liq}is the temperature of liquidus (K); and z is the distance from the meniscus (m).

## 3. Numerical Formulation and Massive Parallelization

## 4. Fuzzy Logic Regulator—FL-BrDSM

## 5. Steel S355 and Real Casting Data

## 6. Results and Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Flick, A.; Stoiber, C. Trends in continuous casting of steel: Yesterday, today and tomorrow. In Proceedings of the METEC InSteelCon, Düsseldorf, Germany, 27 June–1 July 2011; pp. 80–91. [Google Scholar]
- Birat, P.; Chow, C.; Emi, T.; Emling, W.H.; Fastert, H.P.; Fitzel, H.; Flemings, M.C.; Gaye, H.R.; Gilles, H.L.; Glaws, P.C.; et al. The Making, Shaping and Treating of Steel: Casting Volume, 11th ed.; The AISE Steel Foundation: Pittsburgh, PA, USA, 2003; p. 1000. ISBN 978-0-930767-04-4. [Google Scholar]
- Santos, C.A.; Spim, J.A.; Garcia, A. Mathematical modeling and optimization strategies (genetic algorithm and knowledge base) applied to the continuous casting of steel. Eng. Appl. Artif. Intell.
**2003**, 16, 511–527. [Google Scholar] [CrossRef] - Zhemping, J.; Wang, B.; Xie, Z.; Lai, Z. Ant Colony Optimization Based Heat Transfer Coefficient Identification for Secondary Cooling Zone of Continuous Caster. In Proceedings of the IEEE International Conference Industrial Technology, Shenzhen, China, 20–24 March 2007; pp. 558–562. [Google Scholar] [CrossRef]
- Zheng, P.; Guo, J.; Hao, X.-J. Hybrid Strategies for Optimizing Continuous Casting Process of Steel. In Proceedings of the IEEE International Conference Industrial Technology, Hammamet, Tunisia, 8–10 December 2004; pp. 1156–1161. [Google Scholar] [CrossRef]
- Mauder, T.; Novotny, J. Two mathematical approaches for optimal control of the continuous slab casting process. In Proceedings of the Mendel 2010—16th International Conference on Soft Computing, Brno, Czech Republic, 26–28 June 2010; pp. 395–400, ISBN 978-80-214-4120-0. [Google Scholar]
- Ivanova, A.A. Predictive Control of Water Discharge in the Secondary Cooling Zone of a Continuous Caster. Metallurgist
**2013**, 57, 592–599. [Google Scholar] [CrossRef] - Rao, R.V.; Kalyankar, V.D.; Waghmare, G. Parameters optimization of selected casting processes using teaching-learning-based optimization algorithm. Appl. Math. Modell.
**2014**, 38, 5592–5608. [Google Scholar] [CrossRef] - Mosayebidorcheh, S.; Bandpy, M.G. Local and averaged-area analysis of steel slab heat transfer and phase. change in continuous casting process. Appl. Thermal Eng.
**2017**, 118, 724–733. [Google Scholar] [CrossRef] - Mauder, T.; Charvat, P.; Stetina, J.; Klimes, L. Assessment of Basic Approaches to Numerical Modeling of Phase Change Problems-Accuracy, Efficiency, and Parallel Decomposition. J. Heat Transf.
**2017**, 139, 5. [Google Scholar] [CrossRef] - Miettinen, J. IDS Solidification Analysis Package for Steels: User Manual of DOS Version 2.0.0; Helsinki University of Technology: Helsinki, Finland, 1999; p. 22. ISBN 9512246600. [Google Scholar]
- Zhang, J.; Chen, D.F.; Zhang, C.Q.; Wang, S.G.; Hwang, W.S.; Han, M.R. Effects of an even secondary cooling mode on the temperature and stress fields of round billet continuous casting steel. J. Mater. Process. Technol.
**2015**, 222, 315–326. [Google Scholar] [CrossRef] - Javurek, M.; Ladner, P.; Watzinger, J.; Wimmer, P. Secondary cooling: Roll heat transfer during dry casting. In Proceedings of the METEC ESTAD, Düsseldorf, Germany, 15–19 June 2015; p. 8. [Google Scholar]
- Totten, G.E.; Bates, C.E.; Clinton, N.A. Handbook of Quenchants and Quenching Technology; Haddad, M.T., Ed.; ASM International: Materials Park, OH, USA, 1993; p. 507. ISBN 0-87170-448-X. [Google Scholar]
- Ramírez-López, A.; Muñoz-Negrón, D.; Palomar-Pardavé, M.; Romero-Romo, M.A.; Gonzalez-Trejo, J. Heat removal analysis on steel billets and slabs produced by continuous casting using numerical simulation. Int. J. Adv. Manuf. Technol.
**2017**, 93, 1545–1565. [Google Scholar] [CrossRef] [Green Version] - Raudensky, M.; Hnizdil, M.; Hwang, J.Y.; Lee, S.H.; Kim, S.Y. Influence of Water Temperature on The Cooling Intensity of Mist Nozzles in Continuous Casting. Mater. Tehnol.
**2012**, 46, 311–315. [Google Scholar] - Stetina, J.; Mauder, T.; Klimeš, L. Utilization of Nonlinear Model Predictive Control to Secondary Cooling during Dynamic Variations. In Proceedings of the AISTech, Pittsburgh, PA, USA, 16–19 May 2016; p. 14. [Google Scholar]
- Mauder, T.; Sandera, C.; Stetina, J. Optimal control algorithm for continuous casting process by using fuzzy logic. Steel Res. Int.
**2015**, 86, 785–798. [Google Scholar] [CrossRef] - Louhenkilpi, S.; Laine, J.; Miettinen, J.; Vesanen, R. New Continuous Casting and Slab Tracking Simulators for Steel Industry. Mater. Sci. Forum
**2013**, 762, 691–698. [Google Scholar] [CrossRef]

**Figure 1.**Scheme of the continuous casting and mechanical stresses during bending and straightening. 1—tundish; 2—mould; 3—nozzle; 4—cooling circuit; 5—roller.

**Figure 6.**Temperature dependent physical properties for the steel grade S355, (

**a**) thermophysical properties; (

**b**) Mechanical properties.

Numerical Scheme | Computational Time (s) | ||
---|---|---|---|

Coarse mesh | Fine-mesh | Very-fine mesh | |

SE—1 CPU | 11.54 | 887.14 | 54,362.12 |

SE—12 CPU | 135.28 | 1122.13 | 35,124.54 |

ADI—1 CPU | 41.81 | 714.73 | 36,210.41 |

ADI—12 CPU | 58.61 | 824.32 | 32,416.32 |

SE—GPU | 18.67 | 97.72 | 972.49 |

Weight Fraction | Ni | Mn | Mo | Si | Nb | Ti | Cu |

wt% | max 0.300 | 1.400–1.550 | max 0.080 | 0.5 | max 0.060 | max 0.020 | max 0.200 |

Weight Fraction | V | Al | P | C | Cr | S | Ca |

wt% | max 0.020 | 0.020–0.060 | 0.030 | 0.160–0.180 | max 0.200 | 0.020 | 0.002 |

Casting Speed (m/min) | Loop 1 (L/min) | Loop 2 (L/min) | Loop 3 (L/min) | Loop 4 (L/min) | Loop 5 (L/min) | Loop 6 (L/min) |

1.5 | 96.6 | 125.4 | 103.9 | 124.6 | 65.3 | 98.5 |

1.9 | 98.1 | 132.6 | 109.5 | 146.3 | 98.3 | 103.3 |

Casting Speed (m/min) | Loop 7 (L/min) | Loop 8 (L/min) | Loop 9 (L/min) | Loop 10 (L/min) | Loop 11 (L/min) | Loop 12 (L/min) |

1.5 | 32.0 | 65.7 | 22.0 | 34.7 | 31.2 | 43.7 |

1.9 | 46.9 | 85.7 | 26.7 | 58.3 | 99.5 | 106.2 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mauder, T.; Stetina, J.
High Quality Steel Casting by Using Advanced Mathematical Methods. *Metals* **2018**, *8*, 1019.
https://doi.org/10.3390/met8121019

**AMA Style**

Mauder T, Stetina J.
High Quality Steel Casting by Using Advanced Mathematical Methods. *Metals*. 2018; 8(12):1019.
https://doi.org/10.3390/met8121019

**Chicago/Turabian Style**

Mauder, Tomas, and Josef Stetina.
2018. "High Quality Steel Casting by Using Advanced Mathematical Methods" *Metals* 8, no. 12: 1019.
https://doi.org/10.3390/met8121019