# Modeling of the Twin-Roll Casting Process: Transition from Casting to Rolling

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## Abstract

During twin-roll casting, an alloy melt is passing the gap between two counter-rotating rolls, where cooling and solidification leads to the continuous formation of a solid strand. In order to describe this process, a two-phase Eulerian–Eulerian volume-averaging model is presented that accounts for (1) transport and growth of spherical grains within a flowing melt, (2) the formation of a coherent solid network above a specific solid fraction and (3) the viscoplastic flow of the solid network with the interstitial melt during casting and compression. For the considered case of an inoculated Al–4wt%Cu alloy, the process conditions are chosen such that two relatively thick viscoplastic semi-solid shells meet between the rolls, and thus, the material is pressed together and squeezed against the casting direction. The squeezed out material consists of segregated melt and some solid that quickly disappears after melting. It is observed during this study that macrosegregation distributions are inherently connected to the mush deformation that is enforced during the hot rolling process.

## Keywords

Twin-roll casting Solidification Macrosegregation Viscoplasticity Two-phase flow## 1 Introduction

Twin-roll continuous casting is an emerging technology in the casting industry. The process consists of introducing the melt directly into the gap between two counter-rotating rolls, which are cooled so that a solidifying shell forms on each of the moving roll surfaces. Due to the progressively reducing dimensions of the gap between the two rolls, the two shells meet at the so-called kissing point, creating a continuous solidified strip, which is a close-to-final product. The reduced number of operation steps compared to conventional technologies is one of the primary advantages of this technology. As a result, also the operating and energy requirement costs are typically lower [1, 2].

The main feature of twin-roll casting is that, both casting and rolling are merged into one single continuous step. Therefore, a thermo-fluid-mechanical model must be applied for the numerical simulation of this process. Often this is done by using a single phase finite element code originally designed for pure rolling, and treating the liquid as a solid with low viscosity [3, 4]. If a liquid core still exists at the roll nip, the mechanical part of the simulation can be omitted and only solidification and flow should be modeled [5]. However, the viscoplasticity of the semi-solid slurry must explicitly be treated if solidification has already reached the strand center before passing the roll nip. According to Nguyen et al. [6], the solid–liquid mixture can be considered as a viscoplastic continuous solid skeleton saturated with interstitial liquid. Therefore, solid and liquid phases become inherently coupled: if on the one hand, pressing the solid skeleton drives the fluid flow behavior, on the other, the resulting pressure distribution in the interstitial liquid affects the equivalent stress experienced by the solid phase.

In the present paper, a two-phase Eulerian–Eulerian volume-averaging approach is presented to model the transport and growth of equiaxed crystals during solidification. Furthermore, in order to account for mush deformation, a solution algorithm has been developed to include a viscoplastic model when the concentration of solid is above a certain criterion. The model is implemented in OpenFOAM and tested in a pertinent industrial application, where the significance of the viscoplastic behavior of the solid phase cannot be neglected.

## 2 Model Description

The simulation of the twin-roll continuous casting presented in this paper relies on a two-phase Eulerian–Eulerian volume-averaging model. The general approach solves mass, momentum, species and enthalpy conservation equations for each phase. Additionally, a transport equation is also taken into account for the calculation of the number density of grains across the domain. A detailed description of the corresponding governing equations has been made in previous publications [7, 8], so the interested reader can refer to those contributions. For the sake of conciseness, here only the most relevant equations will be highlighted.

Volume-averaged conservation equations

Mass cons. | \( \frac{{\partial g_{i} \rho_{i} }}{\partial t} + \nabla \cdot (g_{i} \rho_{i} {\mathbf{v}}_{i} ) = \mp M_{ls} \) | (1) |

Momentum cons. | \( \frac{{\partial g_{i} \rho_{i} {\mathbf{v}}_{i} }}{\partial t} + \nabla \cdot (g_{i} \rho_{i} {\mathbf{v}}_{i} {\mathbf{v}}_{i} ) = - g_{i} \nabla p + \nabla \cdot g_{i} {\varvec{\uptau}}_{i}^{{\text{eff}}} \mp {\mathbf{U}}_{ls} \) | (2) |

Species cons. | \( \frac{{\partial g_{i} \rho_{i} c_{i} }}{\partial t} + \nabla \cdot (g_{i} \rho_{i} {\mathbf{v}}_{i} c_{i} ) = \mp C_{ls} \) | (3) |

Enthalpy cons. | \( \frac{{\partial g_{i} \rho_{i} h_{i} }}{\partial t} + \nabla \cdot (g_{i} \rho_{i} {\mathbf{v}}_{i} h_{i} ) = - \nabla \cdot {\mathbf{q}}_{i} \mp H_{ls} \) | (4) |

Grain transport | \( \frac{\partial n}{\partial t} + \nabla \cdot ({\mathbf{v}}_{s} n) = 0 \) | (5) |

*A*and

*B*are rheological parameters which, according to Nguyen et al. [6], depend upon the solid fraction as

*m*, are material-dependent quantities. In the present work, they are defined as 6.31 × 10

^{6}Pa·s and 0.213, respectively. The equivalent strain rate is in turn given by

*p*, and the different interphase exchange terms \( M_{\ell s} ,{\mathbf{U}}_{\ell s} ,C_{\ell s} , \) and \( H^{*} \) which are presented in Table 2. The quantities \( v_{r} ,S_{\ell s} ,\varPhi_{\text{imp}} \) and \( K_{\ell s} \) are the crystal growth velocity, specific surface area, impingement factor and momentum exchange coefficient respectively. They have been described in detail in companion papers [7, 8]. So the readers are requested to refer to those papers for further details. The enthalpy exchange term is used to satisfy the condition of thermal equilibrium between phases (with \( h_{c} = 10^{9} \;{\text{W/m}}^{2} /{\text{K}} \)). Lastly, the effective thermal diffusivity for enthalpy in Eq. (16) is equal to 2.49 × 10

^{−5}m

^{2}/s for the liquid and 7.29 × 10

^{−5}m

^{2}/s for the solid.

Exchange terms and other closure laws for the conservation equations

Mass transfer | \( M_{\ell s} = v_{r} S_{\ell s} \rho_{s} \varPhi_{\text{imp}} \) | (12) |

Momentum transfer | \( {\mathbf{U}}_{\ell s} = K_{\ell s} ({\mathbf{v}}_{\ell } - {\mathbf{v}}_{s} ) \) | (13) |

Species transfer | \( C_{\ell s} = c_{s}^{*} M_{\ell s} \) | (14) |

Enthalpy transfer | \( H^{*} = h_{c} (T_{\ell } - T_{s} ) \) | (15) |

Heat flux | \( {\mathbf{q}}_{i} = \alpha_{i} \nabla h_{i} \) | (16) |

## 3 Results and Discussion

Twin-roll continuous casting has been the subject of extensive research due to the increasing interest from industry. It is also an ideal scenario for demonstrating the applicability and robustness of the proposed splitting algorithm which has been developed to account for the viscoplastic regime in an OpenFOAM-based numerical model originally used to address exclusively solidification and transport of equiaxed crystals.

*d*= 5 µm). A heat flux boundary condition is imposed to the roll and strip surfaces, with \( T_{\infty } = 300\;{\text{K}} \). The solid and liquid heat capacities have been defined as \( c_{P,s} = 766\;{\text{J}}/{\text{K}} \) and \( c_{P,\ell } = 1179\;{\text{J}}/{\text{K}} \), whereas the solid and liquid thermal conductivities have been defined as \( \lambda_{s} = 153\;{\text{W}}/{\text{m}}/{\text{K}} \) and \( \lambda_{\ell } = 77\;{\text{W}}/{\text{m}}/{\text{K}} \). The remaining fields not presented in Table 3 are set with homogeneous Neumann boundary conditions.

Boundary conditions for velocities and enthalpy/temperature fields

Solid velocity | Liq. velocity | Enthalpy/temperature | |
---|---|---|---|

Inlet | Pressure inlet | Pressure inlet | 925 K |

Die | Slip | No slip | 925 K |

Roll | Slip | 0.085 rad/s | Heat flux (HTC = 11 kW/m |

Strip | Slip | 0.034 m/s | Heat flux (HTC = 2 W/m |

Outlet | 0.034 m/s | 0.034 m/s | Zero gradient |

As the partly solidified shells are pressed together, material consisting of segregated melt and also some solid is squeezed out against the casting direction. The corresponding solid thus moves into an area where the conditions change from solidification to melting, and therefore the solid gradually disappears. Also, the crystals that are formed further upstream melt by moving into this area of highly segregated liquid. A flow reversal in the vicinity of the roll nip was also reported in Ref. [4]. Similar to the present work, the authors had also considered viscoplasticity of the semi-solid. However, they did not account for segregation and thus melting was not predicted.

## 4 Conclusion

The need for high-quality strips during twin-roll continuous casting makes it imperative for the research community to achieve a better understanding of the physical mechanisms underlying this material process technique. The present model paves the way for such an accomplishment, since it makes it much easier to analyze the influence of different casting parameters in order to build an optimized operation plan. The presented results show that when the partly solidified, viscoplastic shells from both rolls meet at the kissing point, material is squeezed out against casting direction, leading to a local solutally enriched area where crystals melt. The described phenomenon contributes to the formation of a strong centerline macrosegregation.

## Notes

### Acknowledgements

Open access funding was provided by Montanuniversität Leoben. This work was financially supported by the FWF Austrian Science Fund (P28785-N34) which the authors gratefully acknowledge.

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