Simulation of the Deformation of a Continuously Cast Ingot Depending on the Parameters of Mold Flux

We propose a mathematical model of the strained state of steel skin, which enables one to evaluate the initiation of transverse cracks in the ingots. The mathematical model is based on the equations of transient heat conduction, relations between the creep strains and stresses in the steel skin, and the integral balance equations. In our calculations, we use industrial data on the thermal work of the mold: the data of thermocouples mounted in the walls of the mold and the integral heat flux computed according to difference between the temperature of water at the inlet and outlet of the mold. The proposed mathematical model takes into account the thermal properties of mold fluxes, their influence on friction between the ingot and the mold, and the creep properties of cast steels as well as establishes the relationship between the parameters of the process of continuous casting and the appearance of transverse cracks in the steel skin. The results of numerical analysis of the limiting state of steel skin are in good agreement with the industrial data on the initiation of transverse cracks in the ingot.

Transverse cracks may appear in the process of continuous casting of steel both in wide and narrow faces of the slab and in its angular domains. It is customary to distinguish several main causes of the initiation of transverse cracks on the surfaces of products [1,2,3,4]: high concentration of detrimental admixtures in steels, coarse swinging folds, tensile stresses along the ingot, friction in the mold, thermal stresses caused by nonuniform solidification and deformation of the ingot skin in bending or straightening, and high rate of taper of the mold. Independently of the causes and mechanisms of initiation of transverse cracks, the appearance of defects of this kind on the surfaces of blanks may lead to emergency situations or destabilization of the process of casting, rejection of the blanks, and poor quality of the rolled stock.

As shown in [3], it is possible to decrease the amount of transverse cracks by using the optimal mold flux (MF). The detailed analysis of the thermal characteristics of MF can be found in [5], where the authors proposed a mathematical model for the investigation of the dependence of slag-skull thickness on the thermal parameters of the MF. The present work is a continuation of the papers [5, 6]. It is devoted to the problem of simulation of the process of initiation of transverse cracks caused by friction between the ingot and the mold depending on the thermal characteristics of the MF. The main aim of the present work is to perform the numerical analysis of the initiation of transverse cracks based on the industrial and laboratory data.

We propose a mathematical model capable of estimation of the initiation of transverse cracks according to the data of the analysis of inelastic deformation of steel skin in cast blanks. To describe the temperature state of the ingot–mold system, we used the following equations: the integral balance equation (applied to find the drawing force of the ingot), the one-dimensional equation of stationary heat conduction (used to determine the temperature drop between the steel skin and the copper wall of the mold), the one-dimensional equation of transient heat conduction (applied to compute the steel-skin thickness and the temperature field in the skin), and the three-dimensional equation of stationary heat conduction (used to find the heat flux corresponding to the specific conditions of casting, namely, its rate, the type of MF, and the group of steel grades).

Procedure of finding the stress-strain state of steel skin. As shown in [1, 4], the mechanical stresses in the ingot skin can be regarded as one of the causes of initiation of transverse cracks. In the first approximation, to estimate the stress-strain state (SSS) of steel skin, it is necessary to make a series of assumptions:

• 1) the drawing force of ingot F is equal to the friction force F_fr between the steel skin and the mold wall;

• 2) the creep strain rates ξ are uniformly distributed over the thickness of steel skin (flat crossxsection hypothesis) and directed along the axis of casting;

• 3) creep is regarded as the main mechanism of deformation of the steel skin and, hence, the temperature elastic strains can be neglected; and

• 4) it is assumed that there are critical creep strain rates ξ_crit corresponding to the initiation of transverse cracks.

According to the assumption about uniform distribution of the creep strain rates ξ over the thickness of steel skin for each layer of the MF skull at a distance z from the meniscus, we can deduce an integral balance equation for the stresses in the layer σ(x, z), where x is the direction perpendicular to the direction of casting, and z is the direction of casting:

$$ \underset{0}{\overset{d_1(z)}{\int }}\upsigma \left(x,z\right) dx=F(z)/P, $$

(1)

where, σ(x, z) = σ(x, ξ(σ, T)) is the law of creep, F(z) is a force stretching the ingot on a level z from the meniscus, P is the cross-sectional perimeter of the ingot, and d₁(z) is the thickness of the skin on the level z from the meniscus.

The law of creep ξ(σ, T) for steel under the conditions of casting is most often specified in the form of a power dependence in the Norton law. However, as shown in [7, 8], in a wide range of the creep strain rates (including the domain of low stresses and high temperatures), the creep law in the form of hyperbolic sine, namely,

$$ \upxi \left(\upsigma, T\right)=A\exp \left[-{T}_0/T(x)\right]\sinh \left[\upsigma (x)/B\left(T(x)\right)\right] $$

(2)

gives a more adequate agreement with the experimental data. Here, T(x) is the temperature at a specific point with coordinate x on the level z from the meniscus. The constants A, T₀ and the dependence B(T(x)) are the mechanical characteristics of the investigated steels at high temperatures.

By using the integral balance equation (1), we determine the dependence of creep strain rates on the distance from the meniscus ξ(z) provided that the force F(z) was preliminarily found. Further, with the help of relation (2), we establish the distribution of stresses over the thickness of steel skin on the level z from the meniscus.

Modeling of friction between the ingot and the mold. Estimation of the SSS of steel skin is based on finding the friction F_fr(z) between the ingot and the mold. According to our assumptions, the friction force between the ingot and mold on each level z from the meniscus completely determines the force F(z) of drawing of the ingot.

We can estimate friction by using the results obtained in [9], where it is proposed to split friction into the dry τ_s/s (MPa) and liquid τ_s/l components:

$$ {\uptau}_{s/s}={\upmu}_s\left({\uprho}_{\mathrm{slag}}{gh}_0+{\uprho}_{\mathrm{steel}} gz\right); $$

(3)

$$ {\uptau}_{s/l}=\upeta \frac{\left(n+1\right)\left(v-{v}_{\mathrm{m}}\right)}{d_l}-\frac{\left({\uprho}_{\mathrm{slag}}-{\uprho}_{\mathrm{steel}}\right){gd}_1}{n+1}, $$

(4)

where μ _s is the friction coefficient between the ingot and the crystallized MF; ρ_slag is the density of slag, kg/m³; g is the gravitational acceleration, m/sec²; h₀ is the depth of layer of fused MF, m; ρ_steel is the density of steel, kg/m³; η is the viscosity of fused slag, Pa·sec; n is the viscosity index depending on temperature; v is the casting rate, and v_m is the velocity of vibrations of the mold, m/sec; and d _l is the thickness of layer of fused MF, m.

The analysis of dry and liquid friction by using relations (3) and (4) can be performed only if the state of the MF in the gap between the ingot and the mold is known. According to the well-known works [5, 10], the liquid and solid phases of the MF are formed in the gap between the ingot and the mold. Moreover, their mixture undergoes complete crystallization at a certain distance h_0max from the meniscus.

To estimate the state of MF in the gap between the ingot and the mold, we developed a special mathematical model.

Mathematical model of the temperature state of MF in the gap between the ingot and the mold. When MF is supplied into the mold, the mixture is fused and flows into the gap between the ingot and the mold wall.

The problem of heat removal from the ingot can be regarded as the problem of heat conduction across a system formed by the steel skin, fused MF, solidified MF, and a gas gap formed due to the presence of microasperities of the surface of solid MF [1, 10]. The detailed description of this system and the corresponding heat-conduction equations can be found in [5]. In the present work, unlike [5], the equations of stationary heat conduction are solved under several assumptions.

The first assumption is that the total thickness of MF d_MF at any distance from the meniscus z is determined by the thickness of the MF layer on the level of the meniscus for z = 0. The second assumption is the condition under which a gas gap is formed between the ingot and the mold (Fig. 1).

Fig. 1.

Location of one-dimensional computational sections.

If, in the computational scheme, in passing from layer i – 1 to layer i (Fig. 2), the thickness of MF layer (d₂ + d₃)_i–1 is insufficient to balance the heat flux on a level z _i from the meniscus, then, instead of increasing the thickness of the layer on the current level, the required balance is attained as a result of the appearance of a gas gap between the solid MF layer and the mold wall. This condition is formally expressed by the following equations:

Fig. 2.

Average density of the heat flux in the case of casting of low-carbon steels with the use of MF-1 and MF-2.

• in the absence of gas gap, i.e., for Т₃₄ = Т₄₅, we have

$$ \frac{k_2}{d_2}\left({T}_{12}-{T}_{23}\right)={q}_{z_i};\kern0.5em \frac{k_3}{d_3}\left({T}_{23}-{T}_{34}\right)={q}_{z_i};\kern0.5em \frac{k_5}{d_5}\left({T}_{34}-{T}_5\right)={q}_{z_i}, $$

(5)

where d₂ and d₃ are the thicknesses of the layers of fused and crystallized MF, k₂ and k₃ are the thermal conductivities of fused and crystallized MF; T₁₂ is the temperature on the boundary of the steel skin and the mold; T₂₃ is the crystallization temperature of MF; ${q}_{z_i}$ is the heat-flux density on the level z _i from the meniscus; T₃₄ and T₄₅ are, respectively, the temperatures on the boundaries between the crystallized MF and the mold wall in the absence of the gas gap; k₅ is the thermal conductivity of the copper alloy of the mold; and T₅ is the temperature of the mold wall measured by a thermocouple at a distance d₅ from the inner surface of the wall;

• in the presence of a gas gap, if the condition d₂(z _i ) + d₃(z _i ) > d₂(z_i–1) + d₃(z_i–1), where z _i is the coordinate of level of the meniscus for the current layer i in the computational scheme, z_i–1 is the coordinate from the level of the meniscus in the previous layer i – 1 of the computational scheme, is satisfied, then the temperature on the interfaces is determined as follows:

$$ {\displaystyle \begin{array}{ccc}\frac{k_3}{d_3}\left({T}_{23}-{T}_{34}\right)={q}_z;& \frac{k_4}{d_4}\left({T}_{34}-{T}_{45}\right)={q}_z;& \frac{k_5}{d_5}\left({T}_{34}-{T}_5\right)={q}_z,\end{array}} $$

(6)

where k₄ is the thermal conductivity of gas gap, d₂ is its thickness, T₃₄ is the temperature on the boundary between the MF and the gas gap, and T₄₅ is the temperature on the boundary between the gas gap and the mold wall.

As follows from [1], the mean density of the heat flux (as a function of the distance from the meniscus) is given by the formula

$$ q(z)={q}_0-\left({q}_0-{q}_{\mathrm{mean}}\right)\frac{1-{e}^{- kz/v}}{1-\frac{v}{kL}\left(1-{e}^{- kL/v}\right)}, $$

(7)

where L is the height of contact surface between the ingot and the mold (from the meniscus to the lower cut of mold), q_mean is the mean density of the heat flux, q₀ is the density of the heat flux in the domain of the meniscus, and k is a coefficient characterizing the increase in total thermal resistance of the skin with time.

However, under specific casting conditions, only the parameters v, z, and L in relation (7) are known. The other parameters appearing in Eq. (7), namely, q₀, q_mean, and k are unknown and can be found for the investigated steels according to the industrial data, including both the mean density of heat flux computed according to the difference between the temperatures of water at the inlet and outlet of the mold and the flow rate of water (see Fig. 2) and the experimental data on temperature measured by thermocouples in the mold located on levels of 150 (1), 300 (20), and 450 (3) mm from the upper cut of mold (Table 1). The physicochemical parameters of the analyzed MF are presented in Table 2.

Table 1. Experimental Data on Temperature Measured by Thermocouples (y, °C) in the Mold under the Conditions of Casting of Low-Carbon Steels Depending on the Type of MF and Casting Rate (x, m/min)

Table 2. Physicochemical Parameters of the MF

Results of evaluation of the distribution of heat flux along the axis of the mold. The parameters q₀ and k in Eq. (7) are determined from the solution of three-dimensional inverse problem of stationary heat conduction with application of the optimization approach. The inverse problem is constructed according to the following principle: In the three-dimensional model of the mold wall (Fig. 3), on the inner side, we take into account the heat flux and its distribution along the direction of casting according to relation (7). On the other hand, i.e., at the sites of cooling channels, we take into account a boundary condition in the form of contact heat exchange. In the solution of inverse problem, the coefficients q₀ and k in relation (7) are computed to guarantee the agreement between the calculated temperatures at the sites of location of the thermocouples and the experimental data. From the mathematical point of view, this condition takes the following form:

$$ F\left({q}_0,k\right)=\sqrt{\sum \limits_{i=1}^3{\left({T}_i^{\mathrm{exp}}-{T}_i^{\mathrm{com}}\left({q}_0,k\right)\right)}^2}\to \min, $$

(8)

Fig. 3.

Computational scheme of optimization problem: a) for water channels in the conditions of contact heat transfer; b) for the wall on the side of ingot.

where ${T}_i^{\mathrm{exp}}$ and ${T}_i^{\mathrm{com}}$ are, respectively, the experimental and computed temperatures from the solution of the direct three-dimensional problem of stationary heat conduction (see Fig. 3) at three points (i = 1, 2, 3).

We solved the optimization problem for an ingot 1200 × 250 mm in size in the following way: In the first stage, we specify the initial values of q₀ and k. Then we solve the direct problem of stationary heat conduction, determine the tem perature field, and find temperature at three its points, namely, in the zones of mounting of thermocouple. Finally, we compute functional (7) and perform its optimization by the successive solution of direct problems according to the standard optimization algorithms [12].

The solution of the inverse problem makes it possible to determine the optimal values of q₀ and k for the specific MF, steel groups, and casting rates. In Fig. 4, we present the plots of distribution of the heat flux for low-carbon steel as functions of the casting rate and the type of MF. The analysis of these dependences shows that the heat flux observed in the case of using MF-1 in the casting process is weaker than in the presence of MF-2. It was established that the heat flux on the level of the meniscus for MF-2 is higher than for MF-1, depending on the casting rate, by 15–30 rel.%.

Fig. 4.

Distribution of the heat flux depending on the casting rate and the type of applied MF for low-carbon steel: 1) MF-1; 2) MF-2.

The distribution of the heat flux q(z) (see Fig. 4) was used in relations (5) and (6) to determine temperature T _i (z) and the thickness of the layer d _i for all levels z _i from the meniscus.

The assumptions made in the mathematical model of temperature state of the MF in the gap between the ingot and the mold enable us to find the temperature distribution in the ingot–mold system and to determine the thicknesses of the layers of fused slag, crystallized MF, and gas gap (Fig. 5).

Fig. 5.

Thickness of the layers of MF skull in the gap between the ingot and the mold depending on the distance from the meniscus for low-carbon steel: a, b) MF-1 at casting rates of 0.8 and 1.2 m/min; c, d) MF-2 at casting rates of 0.8 and 1.2 m/min.

As follows from Fig. 5, the total thickness of slag skull is different both for MF-1 and MF-2 and for different casting rates. This difference is explained by the thermal properties of the considered MF: their thermal conductivities and the temperatures of crystallization of melted slag [5]. As the casting rate increases, the total thickness of MF decreases. It is also worth noting that, according to the results of calculations, MF-2 partially remains in the liquid phase over the entire distance from the meniscus to lower cut of the mold.

The equations of stationary heat conduction cannot be used for the determination of the temperature of steel skin and its thickness because the latent heat of phase transition in the course of steel solidification is not taken into account in relations (5) and (6) [13]. To determine the thickness of steel skin and its temperature field in the proposed procedure, we apply the equations of transient heat conduction. The description of the equations and the schemes of solution of the problems of transient heat conduction, as applied to finding the thickness of steel skin can be found in [13]. In Fig. 6, we present the results of determination of the thickness of steel skin in the case of casting of low-carbon assortment with the use of MF-1 and MF-2 at different casting rates. The analysis of these data (see Fig. 6) shows that, for MF-1, the thickness of steel skin is smaller than for MF-2. As the casting rate increases, the thickness of steel skin decreases.

Fig. 6.

Thickness of steel skin depending on the distance from the meniscus for low-carbon steel and low-basicity MF-1 (a) and MF-2 (b) for different casting rates.

The results of investigation of the distribution of thickness of fused and crystallized MF in the analyzed model (the quantity h₀ in relation (3)) are used to determine the force of dry friction between the crystallized MF and the copper wall of the mold and the drawing forces of the ingot depending on the distance from the meniscus.

The determination of the drawing force of the ingot enables us to “close” relations (1) and (2) and evaluate, on this basis, the parameters of the SSS: the creep-strain rate as a function of the distance from the meniscus and the distribution of tensile stresses in steel skin caused solely by the action of the drawing force of the ingot F(z). To determine the drawing force, we used the following friction coefficients μ _s (relation (3)) for MF-1 and MF-2: 0.06 and 0.10, respectively [6]. In Fig. 7, we present the distribution of the creep strain rate over the length of ingot for low-carbon steels with application of MF-1 and MF-2.

Fig. 7.

Creep-strain rate in the steel skin for low-carbon steels with MF-1 and MF-2 applied in the process of casting.

The analysis of the SSS of steel skin shows that, in the case of application of the MF considered in the present work, the levels of creep-strain rate are different (see Fig. 7) in the direction of drawing of the ingot from the mold. Note that the difference in the rates is so large that, in this case, it is possible, even without numerical values, to determine the most dangerous situations in which the probability of crack initiation is quite high. Clearly, the application of MF-1 from the viewpoint of crack initiation is worse than the application of MF-2.

The comparison of the plots in Fig. 7 with the industrial data on the initiation of transverse cracks reveals their good agreement. The industrial data (rejection of slabs according to the amount of transverse cracks) show that the number of rejected slabs for MF-2 is much lower than for MF-1. Thus, in analyzing the influence of MF on the hazard of initiation of transverse cracks in a continuously cast blank, the creep-strain rate of an ingot in the mold in the direction of its drawing can be used as one of the parameters in estimating the hazard of crack initiation in the ingots.

Conclusions. We develop a mathematical model of heat transfer and slag solidification between the ingot and the mold, which enables one to determine the thickness of slag interlayer and estimate the parameters of cooling of the ingot depending on the casting rate, steel grade, and type of MF.

On the basis of mathematical simulations and the analysis of the industrial data, we determine the distribution of the heat flux for different casting rates of steels of low-carbon assortment with the use of different MF. It is established that the heat flow on the level of the meniscus may differ by 15–30 rel.% depending on the MF and the casting rate.

In addition, we determine the influence of the casting rate and the thermal properties of MF on the slag-skull thickness: as the casting rate increases, the thickness of the MF layer decreases.

We also propose a criterion of initiation of transverse cracks based on the mathematical model of heat transfer in the ingot–mold system. This criterion enables us to estimate the probability of initiation of transverse cracks in the ingots. It is shown that the results of evaluation of this criterion for steels of the low-carbon assortment subjected to casting with two industrial MF are in good agreement with the experimental data on the rejection of slabs according to the presence of defect in the form of transverse cracks.