Introduction

One of the most important technological processes in modern metallurgy is continuous casting. During the cooling and solidification of a slab in a continuous casting machine (CCM), metal shrinkage occurs, resulting in the formation of a shrinkage cavity in the upper part of the final slab. Of particular interest is the development of a method for calculating the shrinkage of a continuously cast slab in the mold and the secondary cooling zone (SCZ) of a CCM, as well as determining the slab dimensions with shrinkage taken into account. A number of mathematical models describing the processes of slab cooling and metal shrinkage have been developed. Research on this topic began in the 1970s. For example, study [1] was devoted to calculating shrinkage during solidification of a billet with a circular cross-section. Subsequently, the developed models became more sophisticated, incorporating hydrodynamic phenomena in the liquid phase [2], more detailed descriptions of slab bulging [3], phase transformations during casting [4], thermomechanical phenomena [5], heat transfer in the secondary cooling zone [6], and other factors.

The development of a predictive shrinkage model would make it possible to account for the influence of shrinkage on technological parameters such as mold taper and the roller gap in the SCZ. In study [7], an attempt was made to predict the “ideal” mold taper using a two-dimensional mathematical model. Attempts were also made to develop a model that takes into account the carbon content in steel [8].

In Russian research, particular attention has been given to the works of Dyudkin D.A. [9] and Samoylovich Yu.A. and Kabakov Z.K. [10; 11], who developed a universal model describing the cooling and solidification of a continuously cast slab. This model subsequently formed the basis for numerous further developments, including models of slab surface layer formation [12], continuous slab deformation [13], and the secondary cooling zone (SCZ) system of the CCM [14; 15], among others.

Most proposed models assume that the thermal properties of the metal are independent of temperature. In many cases, these properties are taken as constant for both the solid and liquid phases. When reliable temperature-dependent data are unavailable, the properties are typically estimated by linear interpolation. Moreover, the calculated shrinkage parameters strongly depend on the selected numerical method: study [16] shows that results obtained using different models for identical input data may differ by more than a factor of two

Thus, the existing models of shrinkage processes are not universal and remain imperfect, since they do not account for all features of slab solidification and cooling in a CCM. Therefore, there is a need to develop a more comprehensive mathematical model of shrinkage processes in continuously cast slabs.

Fundamental theoretical provisions

The mathematical model of cooling and solidification of a continuously cast slab is based on the solution of the transient heat conduction equation:

where ceff is the effective heat capacity of steel; ρ is the steel density; λ is the thermal conductivity of the melt; B is half of the slab thickness, m; h is the slab height, m; v is the slab withdrawal rate along the vertical axis, m/min; Θ is the heat source of superheated metal in the region influenced by the casting jet.

where cm is the molecular heat capacity of the alloy; Тs and Тl are the solidus and liquidus temperatures, respectively; L is the latent heat of solidification of the metal in the solid–liquid (two-phase) zone; \(\frac{{d\psi }}{{dT}}\) is the rate of phase transformation during equilibrium solidification of the binary alloy.

Since metal solidification depends on the percentage carbon content, establishing the temperature dependence ceff (T) requires consideration of the Fe–C phase diagram, particularly its high-temperature part (Fig. 1).

The diagram shown in Fig. 1 identifies three steel groups: I – [% С] ≤ 0.1; II – 0.1 < [% С] ≤ 0.16; III – 0.16 < [% C] ≤ 0.5. For each group, the temperature dependence ceff (T) was determined, where:

where Ll – δ is the latent heat of the liquid → δ-ferrite transformation; Lδ – γ is the latent heat of the δ-ferrite → austenite transformation; ψδ is the volume fraction of δ-ferrite in the elementary volume of the “liquid + δ-ferrite” mixture; ψγ is the volume fraction of austenite in the “δ-ferrite + austenite” mixture; \(\frac{{d{\psi _\delta }}}{{dT}}\) and \(\frac{{d{\psi _\gamma }}}{{dT}}\) are the rates of formation of δ-ferrite and austenite, respectively.

where δ(T) – is the Dirac delta function, which was approximated by the expression \(\delta (x) \approx \frac{1}{{a\sqrt \pi  }}{e^{ - {{\left( {\frac{x}{a}} \right)}^2}}};\)

Similarly, the temperature dependences of the effective thermal conductivity λeff (T) and the density ρ(T) were determined.

The calculation of the linear shrinkage coefficient αl (T) was carried out based on the obtained values of the thermal parameters. Since the temperature dependence of the specific volume V(T) in the Fe–C system is strongly influenced by the carbon concentration, it can be concluded that a similar dependence also applies to linear shrinkage. In the proposed model, the coefficient αl (T) is calculated using the expression for volumetric shrinkage given in [17]:

where V(T) is the temperature-dependent specific volume of the alloy; \(\frac{{dV(T)}}{{dT}}\) is the temperature derivative of the specific volume.

The calculated values of the volumetric shrinkage coefficient αV  (T) are subsequently used to determine the shrinkage cavity depth [18].

To solve equation (1), the finite difference method (FDM) was applied using an explicit scheme for approximating the partial derivatives, with the introduction of fictitious nodes that allow more accurate approximation of boundary derivatives with respect to x and y. The computational domain represents one quarter of the slab cross-section (Fig. 2). The number of nodes in the region 0 ≤ x ≤ A is taken as N, and in the region 0 ≤ x ≤ B as M.

At the initial time, the temperature is assumed to be uniform and equal to the liquidus temperature:

The boundary conditions on boundaries Г3 and Г4 are:

The boundary conditions for the cooling zones along boundary Γ1 are defined as follows:

– in the mold:

where αm is the heat transfer coefficient in the mold; \(q =  - \lambda \frac{{\partial T}}{{\partial w}},\) where w = x, y is the heat flux from one face of the slab;

– in the i-th SCZ section:

– air cooling:

where αl is the heat transfer coefficient in the i-th SCZ section; Тs is the slab surface temperature; Тw is the temperature of the cooling water in the mold; Тm is the temperature of the working (copper) wall of the mold; Тam is the ambient temperature; σl is the radiation coefficient; αl is the effective heat transfer coefficient during air cooling; αc is the convective heat transfer coefficient Boundary conditions on Г2 are specified in a similar manner.

The procedure for calculating linear shrinkage is as follows. The calculation is performed for one quarter of the slab cross-section, and the solidus isotherm is taken as the boundary of complete solidification. In [19], an expression was obtained for the relative strain rate of the solid shell:

where ξ is the thickness of the solid shell; αl (Т) is the linear shrinkage coefficient of steel; \(\dot T\) is the temperature change rate at point x at time t.

After integrating this expression over the width of the face, the strain rate expressions for the wide face (ηwf ) and narrow face (ηnf ) of the slab are obtained. The total deformation of the faces from the onset of solidification is then determined:

where ξшг and ξуг are the thicknesses of the solidified layer at the wide and narrow faces, respectively, mm; αl (T) is the linear shrinkage coefficient, °C–1; \(\dot T\) is the rate of temperature decrease/increase at point x at time t, °C/s.

Then, the slab thickness and width during solidification are determined as follows

Verification

The model was implemented as a computer program. The following input data were used in the simulations: slab thickness B = 0.2 m, slab height h = 6.5 m, initial temperature T0 = 1520 °С, ambient temperature Тam = 30 °С, liquidus temperature Тl = 1500 °С, solidus temperature Тs = 1450 °С, number of nodes across the thickness M = 20, number of nodes across the height N = 2500, casting rate v = 0.4 m/min, and cross-section 200×1200 mm. The mold length, SCZ section lengths, and the heat transfer coefficients for the individual sections are given in Table 1.

Verification was performed by comparing the calculated shrinkage cavity depth with the experimental data reported in [20]. According to [20], the mean cavity depth for a sample of 26 slabs is 0.353 m. In turn, the shrinkage cavity depth obtained in the simulation was 0.35 m (Fig. 3).

During verification, the dependence of the relative modeling error on the number of grid nodes along the height N and thickness M was also identified (see Tables 2 and 3).

Tables 2 and 3 show that modeling accuracy increases with increasing numbers of nodes along the height and thickness, reaching an acceptable level of <5 % at N = 2500 and M = 20.

Conclusions

A mathematical model of the shrinkage process in a continuously cast slab has been developed. Its distinctive feature is the incorporation of temperature-dependent thermal properties of the metal and technological casting parameters.

Model verification confirmed the agreement between the simulation results, theoretical calculations, and experimental data.

The modeling accuracy increases with increasing numbers of computational grid nodes along the height N and thickness M, with M having a stronger effect on the modeling accuracy.

The proposed model can be used to improve continuous casting technology.