Friction Stress Analysis of Slag Film in Mold of Medium-Carbon Special Steel Square Billet

Abstract

Non-uniform friction and lubrication are the key factors affecting the surface quality of the casting billet. Based on the three-layer structure of the casting powder in the mold, the frictional stress in the mold was calculated and analyzed by using the relationship between the frictional stress and the thickness and viscosity of the liquid slag film, and the lubrication state between the cast billet and the mold was evaluated. Based on the actual production data of 40Mn2 steel and combined with the numerical simulation results of the solidification and shrinkage process of the molten steel in the mold by ANSYS 2022 R1 software, the frictional stress on the cast billet in the mold was calculated. It was found that within the range of 44~300 mm from the meniscus, the friction between the cast billet and the mold was mainly liquid friction, and the friction stress value increased from 0 to 145 KPa. Within 300–720 mm from the meniscus, the billet shell is in direct contact with the mold. The friction between the cast billet and the mold is mainly solid-state friction, and the friction stress value increases from 10.6 KPa to 26.6 KPa. It indicates that the excessive frictional stress inside the mold causes poor lubrication of the cast billet. By reducing the taper of the mold and optimizing the physical and chemical properties of the protective powder, within the range of 44~550 mm from the meniscus, the friction between the cast billet and the mold is mainly liquid friction, and the friction stress value varies within the range of 0–200 Pa. It reduces the frictional stress inside the mold, improves the lubrication between the billet shell and the mold, and completely solves the problem of mesh cracks on the surface of 40Mn2 steel cast billets.

1. Introduction

Medium-carbon special steel is a type of carbon steel with a carbon content of 0.25% to 0.60%. It exhibits relatively good plasticity, toughness, and strength, while also possessing excellent machinability. Compared with low-carbon steel, medium-carbon special steel can be used after heat treatment or directly as cold-rolled or hot-rolled materials without heat treatment. Therefore, it is often used to manufacture high-strength components, such as air compressors, worm gears, machine tool bearings, and components requiring high wear resistance.

However, its hardenability and weldability are poor, and the surface of the casting slab is prone to cracking. Mold flux plays a very critical role in the quality of the casting slab, especially for steel grades with high crack sensitivity [1] According to the solidification characteristics of 40Mn2 medium-carbon special steel, it is necessary to control the lubrication and heat transfer of the mold. This objective can be readily achieved by adjusting the mold process parameters and optimizing the physical and chemical properties of the casting powder [2]. Gan and colleagues [3] established a mold solidification experimental device to analyze the frictional resistance in the mold. The results show that the variation law of frictional force is consistent with the variation law of the mold vibration velocity. During the positive slip period, the frictional resistance of the mold is the highest, and the initial solidified shell is subjected to the maximum tensile stress. During the negative slip period, the initial solidified shell is subjected to the maximum compressive stress. During the negative slip period, the compressive stress of the initial shell is the greatest. Cai and colleagues [4] found that there exist solid slag films and liquid slag films between the mold and the casting billet. Below the meniscus, in regions where the slag film is completely liquid or coexists in solid–liquid states, the frictional force between the casting billet and the mold is liquid friction; in regions where the surface temperature of the casting billet is lower than the solidification temperature of the casting powder, the slag film is completely solid, and the frictional force between the casting billet and the mold is solid friction at this time. Meng and colleagues [5] put forward a theoretical calculation method for both slag consumption and maximum liquid friction force of liquid slag during stable flow under non-sinusoidal vibration conditions in a continuous casting mold. This method provides a basis for in-depth research on the lubrication mechanism within the mold, prevention and control of surface cracks, improvement of heat transfer conditions, and optimization of vibration parameters and the physical and chemical properties of casting powder. Mizukami and colleagues [6] proposed a calculation model for the frictional force on the surface of the solidified shell. By determining the distribution of the slag channel between the mold and the solidified shell, the model quantitatively calculates the distribution of liquid and solid frictional stress on the surface of the solidified shell within the mold. Additionally, he pointed out that in the upper part of the mold, the surface of the solidified shell is dominated by liquid friction, while solid friction prevails in other positions. Valentin and colleagues [7] found that lubrication is influenced by casting powder, frictional force, and mold vibration in the mold. By measuring the frictional forces of different casting billets, the relationship among friction, steel grades, and casting powder can be evaluated. Based on statistical analysis, Däcker and colleagues [8] adopted a mold with concave narrow faces to compensate for the edge shrinkage of the casting slab and adjusted the mold taper, which significantly reduced the surface cracks of peritectic steel. Wang and colleagues [9] proposed a mold vibration and optimization method for the online measurement of mold friction, design of negative oscillation parameters, and evaluation of slag consumption. Meanwhile, the influence of mold vibration on friction and lubrication was studied. Odagaki and colleagues [10] evaluated lubrication and heat transfer by measuring the frictional force in the mold and found that the higher the solidification temperature of the casting powder, the greater the frictional force in the mold; the frictional force is directly proportional to the velocity of the slab relative to the mold. In the study of surface cracks in the concave area of a wide slab, Niu and colleagues [11] found that appropriate mold taper can constrain the solidification shrinkage of the shell. The temperature distribution on the surface of the solidified shell in the mold serves as the basis for calculating the frictional force. Meanwhile, the solidification and shrinkage of the shell, along with the distribution of the casting powder slag film and air gap, collectively determine the heat transfer within the mold [12]. Therefore, establishing a reasonable and accurate numerical model for heat transfer in the mold zone of continuous casting billets is a key link in studying the friction behavior of the solidified shell surface.

This paper quantitatively analyzes the influence of friction on the network cracks of medium-carbon special steel by calculating the frictional stress in the mold. Meanwhile, ANSYS software is used to numerically simulate the solidification shrinkage process of the casting slab. By adjusting continuous casting process parameters such as mold taper and optimizing the physical and chemical properties of mold flux, the frictional stress between the casting slab and the mold is improved, heat transfer is enhanced, and lubrication is optimized, so as to achieve the purpose of improving the network cracks on the surface of the casting slab.

2. Analysis of Quality Defects of Casting Billet

2.1. Process Parameter

The medium-carbon special steel 40Mn2 produced by a factory was taken as the research object, and its main components are shown in

Table 1. Main chemical composition of 40Mn2 steel.

Ingredient	C	Si	Mn	P	S	Cr	V	Al	Cu
content/%	0.3~0.4	0.5~0.6	1~2	0.014	0.05	0.17	0.127	0.012	0.071

.

2.2. Defects of Casting Billet

During the continuous casting production of 40Mn2 steel, quality defects such as surface network cracks can easily occur, which seriously affect the quality of the casting slab. The microscopic morphology of the surface network cracks on the casting slab is shown in Figure 1.

The cross-section of the sample crack was analyzed by long-term area scanning and fixed-point scanning using a scanning electron microscope (SEM) combined with an energy-dispersive spectrometer (EDS). Points were taken around the network crack, and a component analysis was performed using the EDS. The sampling positions and data results are shown in Figure 2 and

Table 2. Results of EDS component data at selected points.

Element	O	Al	Si	Ca	Cr	Fe	Cu
Weight	6.24	0.14	0.28	0.18	1.01	39.24	1.23
Atom	7.15	0.10	0.18	0.08	0.36	12.88	0.35
Compound	6.24	0.14	0.28	0.18	1.01	39.24	1.23

.

As can be seen from Table 2, the content of Cu and Cr elements at the crack is significantly higher than the normal content of Cu and Cr elements in medium-carbon special steel, especially for the Cu element, which exceeds the normal content by about 20 times, indicating that there is Cu enrichment, penetration, and a Cr segregation phenomenon at the crack. Meanwhile, the high O content is caused by oxides. This indicates that under the action of external stress, the friction between the inner wall of the mold and the casting billet is too large, resulting in severe wear of the Cr coating on the mold wall.

3. Simulation of Solidification Shrinkage Process of Medium-Carbon Special Steel

In order to study the surface temperature of the casting billet and the air gap between the slab and the mold, ANSYS software was used to simulate and analyze the solidification shrinkage process of the bloom in the mold. A solidification heat transfer model was established based on the heat transfer equation and mechanical equation. Meanwhile, while ensuring consistency with the actual production process, the solidification heat transfer model of the casting billet was simplified, and the following assumptions were made [13,14,15,16,17,18,19]:

(1)

Since the longitudinal heat transfer of the casting billet is far less than its transverse heat transfer, the longitudinal heat transfer was ignored, and a two-dimensional steady-state model was established to simulate the solidification shrinkage process of the billet in the mold;

(2)

The influence of mold oscillation on the heat transfer and solidification of the casting billet was ignored;

(3)

Ignoring the longitudinal deformation of the billet, the lateral deformation of the billet was assumed to be a generalized plane;

(4)

The convective heat transfer process in the liquid and two-phase regions was replaced with equivalent heat conduction;

(5)

An elastoplastic model was used to characterize the mechanical characteristics of the billet at a high temperature;

(6)

The initial temperature of the liquid steel remained the same in the calculation area;

(7)

The material obeyed the Von Mises yield criterion;

(8)

Because the cross section of the casting blank is axisymmetric, a quarter of its cross section was used as the analysis model to simplify the calculation work.

3.1. The Establishment of Mathematical Model

(1)

Initial conditions

The initial temperature of the model is the temperature of the molten steel when it first enters the mold, which is the pouring temperature of the molten steel in normal continuous casting production. The pouring temperature used in this paper is 1520 °C. The other parameters required for the simulation are shown in

Table 3. Other process parameters for simulation.

Item	Value	Unit
Casting section	530 × 410	mm
Casting temperature	1520	°C
Cooling water inlet temperature	30	°C
Cooling water outlet temperature	35	°C
Cooling water velocity	8	m/s
Water tank depth	7	mm
Mould effective length	720	mm
Copper wall thickness	21	mm
Cooling water volume	210	W/(m3·h−1)
Taper	1.45	%·m−1
Solidus Temperatures	1417	°C
Liquidus Temperatures	1492	°C

.

(2)

Boundary conditions

Transverse heat transfer of the casting billet is the second type of boundary condition, and the heat flux is calculated by Savage’s formula:

$$q=1.65-0.096\sqrt{t}$$

(1)

In Equation (1),

$q$—Thermal flux, MW·m⁻²;

$t$—The time the liquid steel is in the mold, s.

The heat transfer process between the casting billet and the mold is a relatively complex process. Since all the heat from the mold is absorbed by the cooling water, the heat transfer process can be represented by an equivalent heat flux density.

(3)

Basic equations of the model

A two-dimensional steady-state heat transfer model was established based on the solidification heat transfer problem of the billet in the mold, and its heat transfer differential equation is as follows:

$$\rho c{\displaystyle \frac{\partial T}{\partial t}}={\displaystyle \frac{\partial }{\partial x}}\left(\lambda {\displaystyle \frac{\partial t}{\partial x}}\right)+\left(\lambda {\displaystyle \frac{\partial t}{\partial y}}\right)+q_v$$

(2)

In Equation (2),

$\rho$—Molten steel density (kg/m³);

$c$—Specific heat capacity (J/kg·K);

$\lambda$—Thermal conductivity (W/m °C);

$q_v$—Internal heat source (W/m³).

The strain of the billet is divided into three types: thermal strain, plastic strain, and elastic strain. The relationship among these three types of strains is as follows:

$$d{\epsilon }_{ij}=d{\epsilon }_{ij}^T+d{\epsilon }_{ij}^p+d{\epsilon }_{ij}^e$$

(3)

In Equation (3),

$d{\epsilon }_{ij}$—Total strain variable;

$d{\epsilon }_{ij}^T$—Thermal strain variable;

$d{\epsilon }_{ij}^p$—Plastic strain variable;

$d{\epsilon }_{ij}^e$—Elastic strain variable.

Metallic materials obey the Von Mises yield criterion, and their yield surfaces can be expressed as follows:

$$\overline{\sigma }-{\tau }_s={\left({\displaystyle \frac{1}{2}}{\sigma }_{ij}^{\prime }{\sigma }_{ij}^{\prime }\right)}^{{\displaystyle \frac{1}{2}}}-{\tau }_s\left(\overline{{\epsilon }_{ij}},T\right)=0$$

(4)

In Equation (4),

$T$—Temperature;

${\tau }_s$—Shear yield stress;

${\sigma }_{ij}^{\prime }$—Stress deviator vector.

${\tau }_s$ can be expressed by the following formula:

$${\tau }_s={\tau }_0\left({\overline{\epsilon }}^p\right)-{\tau }_1\left(T\right)$$

(5)

In Equation (5), ${\overline{\epsilon }}^p$ is the equivalent plastic strain, which can be expressed by the following formula:

$$d{\overline{\epsilon }}^p=\sqrt{{\displaystyle \frac{2}{3}}d{\epsilon }_{ij}^{p}d{\epsilon }_{ij}^p}$$

$$d{\epsilon }_{ij}^p=d\lambda {\displaystyle \frac{\partial \overline{\sigma }}{\partial {\sigma }_{ij}^{\prime }}}$$

In the formula, $d\lambda$—scale-up factor.

Through the formulas of elastic mechanics and the above equations, it can be concluded that

$$d{\sigma }_{ij}=2Gd{\sigma }_{ij}+{\displaystyle \frac{v}{1-2v}}d{\epsilon }_{kk}{\delta }_{ij}-{\displaystyle \frac{1+v}{1-2v}}\alpha dT{\delta }_{ij}-{\sigma }_{ij}^{\prime }{\displaystyle \frac{{\sigma }_{mn}d{\epsilon }_{mn}}{S}}-2G{\displaystyle \frac{Q}{S}}{\sigma }_{ij}^{\prime }$$

(6)

In Equation (6), mn represents the region related to plasticity, and the governing equation for the plastic behavior of the model can be simplified as follows:

$$d{\sigma }_{ij}=D^{p}d\epsilon -2G{\sigma }^{\prime }{\displaystyle \frac{Q}{S}}dT$$

The parameters in the equation can be expressed as follows:

$$S={\displaystyle \frac{2}{3}}{\overline{\sigma }}^2{\displaystyle \frac{H^{\prime }}{3G}}+1$$

$$Q={\displaystyle \frac{\beta \overline{\sigma }}{\partial {\overline{\epsilon }}^P}}$$

$$H^{\prime }={\displaystyle \frac{\partial \overline{\sigma }}{\partial \overline{\epsilon }p}}$$

$$\overline{\sigma }=\sqrt{{\displaystyle \frac{3}{2}}{\sigma }_{ij}^{\prime }{\sigma }_{ij}^{\prime }}$$

$$\beta ={\displaystyle \frac{\partial {\tau }_1}{\partial T}}$$

(4)

Finite element model

According to the listed basic assumptions of the model, the solidification and heat transfer process of the bloom in the mold can be simulated by a two-dimensional steady-state heat conduction method. The solidification and heat transfer process of the billet from the meniscus to the mold outlet was studied by a two-dimensional slicing method. Using the two-dimensional thermal-mechanical coupling method, the temperature of the billet from the meniscus to the mold outlet, the thickness of the shell, and the variation law of the air gap thickness can be obtained. The two-dimensional heat transfer mathematical model established in this study is a heat conduction model with heat sources, and there is no additional heat source. The latent heat released by the phase change in molten steel is processed by the enthalpy method, and the equivalent heat capacity method in the mathematical model of billet heat transfer. The equivalent heat capacity method provided by the ANSYS heat transfer module is used for analysis in this study.

In this paper, the ANSYS software is used to establish a solid model. The length of the mold is 800 mm, the thickness of the copper wall is 21 mm, and the height of the molten steel level is 720 mm. The element type adopts the PLANE55 element, and the MAPPED method is used to divide the model grid. The billet was divided into 10 regions to ensure the accuracy of the model and optimize the calculation amount of the model. Specifically, the element spacing for the rounded corners is 0.5 mm, the two-phase zone is 1 mm, the area away from the corner is 2 mm, the surface central area is 4 mm, and the inner corner part is divided by the FREED method. The finite element model is shown in Figure 3.

(5)

Verification of model accuracy

To prevent the simulation results from deviating significantly from actual production, some field-measured data were compared with the simulation results to verify the accuracy of the model calculation results. The surface temperature at the center of the wide face of the casting billet when it exited the mold was measured on-site using an infrared thermometer.

Table 4. Contrast of measured temperature and calculate temperature.

The Casting Billet Exits the Mold	Center Temperature of the Wide Face/°C
Measured temperature	1050
Simulated temperature	1045.57
Deviation	0.42%

shows the comparison results between the billet temperature calculated by the mathematical model and the measured temperature.

3.2. Analysis of Results

(1)

Analysis of temperature field of casting billet

In order to analyze the temperature distribution law of the casting surface in the direction of drawing, Figure 4 shows the temperature change curve of the corner and the central feature point of the surface from the meniscus to the mold outlet.

Since the temperature change trends at the center of the wide face and the center of the narrow face are basically consistent, the temperature change in the characteristic point at the center of the wide face of the casting slab was taken as the temperature change at the center of the surface. It can be seen from Figure 4 that near the meniscus, the change trends of the corner and the surface center are consistent, but as the distance from the meniscus increases, the temperatures of both the corner and the surface center decrease, and the temperature of the corner is lower than that of the surface center. At the mold outlet, the temperature at the slab corner is 753 °C, and the temperature at the surface center is 865 °C. At a distance of approximately 300 mm from the meniscus, the temperature drop rate of both the corner and the surface center significantly accelerates, possibly indicating direct contact between the billet and the mold. Meanwhile, the casting powder is near the transition temperature, causing the frictional force between the billet and the mold to increase. When the frictional force on the shell exceeds the shell’s bearing capacity, network cracks appear on the billet surface. From this, it can be seen that the taper design of the mold is unreasonable.

(2)

Air gap analysis between casting billet and mold

Figure 5 shows the change curve of the air gap of the characteristic points on each surface of the casting blank as the distance from the meniscus increases.

As shown in Figure 5, the air gap between the slab corner and the mold is the largest, followed by that on the narrow face, and the smallest is on the wide face. With the increase in the distance from the meniscus, the air gap between the slab and the mold first increases and then decreases. At a distance of approximately 300 mm from the meniscus, the air gaps between the corner center, wide face center, and narrow face center and the mold disappear successively. The possible reason is that the reduction in the inner cavity size of the mold at this point has exceeded the shrinkage of the slab, leading to direct contact between the mold and the slab, which increases the frictional stress between them. When the frictional stress exceeds the bearing capacity of the shell, network cracks are likely to form on the shell surface, and in severe cases, there may even be a risk of breakout. Therefore, the mold taper should be reduced to increase the air gap between the slab and the mold, ensuring a close fit between them while avoiding solid-state friction.

4. Calculation of Friction Stress in Mold

The form of frictional stress between the mold and the solidified shell depends on the contact state between the casting slab and the mold flux. When the mold flux between the solidified shell and the mold exists in a liquid state, liquid friction occurs in the mold; when the mold flux between the solidified shell and the mold exists in a solid state, solid friction occurs in the mold [20,21,22].

For the friction stress on the surface of the billet shell in the mold, this study adopts the mathematical model proposed by Mizukami [6] as the basis for calculation, as shown in Equations (7) and (8). It is assumed that the structure of the three layers of slag film in the mold is as follows: the liquid slag layer close to one side of the casting billet surface, the glass layer in contact with the copper wall of the mold, and the crystal layer between the liquid slag layer and the glass layer [23,24]. At the same time, considering the influence of mold oscillation speed and temperature on the viscosity of the casting powder, the liquid friction on the surface of the casting billet can be calculated more accurately after some assumptions and simplification.

The liquid friction stress between the casting billet and the mold depends on the thickness of the liquid slag layer and the viscosity of the casting powder, which can be expressed by Equation (7):

$$f_1=\eta {\displaystyle \frac{V_m-V_c}{d_l}}$$

(7)

In Equation (7),

$\eta$—Viscosity of casting powder;

$V_m$—Mold oscillation speed;

$V_c$—Casting speed;

$d_l$—Thickness of liquid slag film.

Assuming that the air gap between the mold and the casting billet is filled with slag film, and the temperature in the slag film changes linearly, the thickness of the liquid slag film can be calculated according to the temperature distribution and the melting temperature of the casting powder.

$$\begin{array}{l}{d}_l={\displaystyle \frac{T_{surf}-T_{melt}}{R_H}}\\ R_H={\displaystyle \frac{T_{surf}-T_{mould}}{G_H}}\end{array}$$

In the above Equation,

$R_H$—Temperature gradient in slag film at distance H below meniscus;

$T_{surf}$—Surface temperature of solidified billet;

$T_{mould}$—Temperature of hot surface of mold copper plate;

$G_H$—Width of the air gap at distance H below the meniscus;

$T_{melt}$—Transition temperature of casting powder.

For solid friction stress, it often depends on the exact solid friction coefficient. Yu [25] experimentally tested the solid friction coefficient of the protective slag at different temperatures, and found that when the temperature increased from 25 °C to 600 °C, the friction coefficient of the crystalline slag would increase from 0.98 to 1.52, but if the temperature continued to increase to 800 °C, the friction coefficient would decrease to 0.75. Therefore, in this study, the solid-state friction coefficient is 0.5. At the same time, it is considered that when the surface temperature of the billet is lower than the transition temperature of the casting powder, the friction of the billet shell in the mold is mainly solid friction.

Solid friction stress is calculated as in Equation (8):

$$\begin{array}{l}{f}_s={\eta }_{s}H\\ H=7.4h\end{array}$$

(8)

In Equation (8),

${\eta }_s$—Friction coefficient between the copper wall of the mold and the solid slag film;

$H$—Static pressure of molten steel;

$h$—Depth of liquid steel.

It should be noted that the hot surface temperature of the mold copper plate is related to actual process conditions, and the hot surface temperature varies at different positions within the mold; for the sake of simplifying calculations, the hot surface temperature of the mold copper plate is assumed to be 300 °C. In the ANSYS numerical simulation, the influence of mold vibration on heat transfer and solidification is ignored, and the effect of mold vibration on frictional stress is not considered in the calculations. However, the ANSYS simulation analysis shows that the variation trend of the slab vertical stress is the same as that of the frictional stress, which supports the reliability of the calculation results.

The surface temperature $T_{surf}$ and gap width $G_H$ of the casting billet were obtained from the numerical simulation of ANSYS solidification shrinkage mentioned above. The continuous casting process parameters and other calculation parameters used in the calculation are shown in

Table 5. Continuous casting process parameters.

Item	Unit	Parameter
Mold copper tube length	mm	800
Cast section	mm	410 × 530
Mold amplitude	mm	2
Mold oscillation frequency	frequency/min	110
Mold taper Angle	%·m−1	1.6
Pouring temperature	°C	1520

and

Table 6. Other calculation parameters.

Item	Unit	Parameter
Casting speed	m/min	0.45
Melting temperature of casting powder	°C	1211
Transition temperature of casting powder	°C	1150
Viscosity of mold slag	Pa·s	0.204
Temperature of hot surface of mold copper plate	°C	300
Friction coefficient between mold copper wall and solid slag film		0.5

.

Equations (7) and (8) can be used to calculate the change in friction stress in the mold with the distance from the meniscus, as shown in Figure 6.

It can be seen from Figure 6 that the solid friction stress between the casting billet and the mold increases linearly with the increase in the distance from the meniscus. The friction stress in the mold is mainly liquid friction stress because the surface temperature of the casting billet is higher than the transition temperature of the casting powder when the casting billet is 0~250 mm away from the meniscus. At the same time, the friction stress from 0~250 mm away from the meniscus increases slowly, indicating that the mold lubrication in this area is good. The liquid friction stress of the billet increases sharply near 300 mm from the meniscus. The reason is that near 300 mm, the thickness of the liquid slag film becomes sharply thinner, or the type of friction directly changes from liquid friction to solid friction. This is basically the same as the variations in the shrinkage of characteristic points on each surface of the billet described in Figure 5.

Combined with Figure 4 and Figure 6, it was found that the casting billet surface temperature drops faster at 300 mm away from the meniscus of the mold, and the casting powder is at the transition temperature. Therefore, the friction stress in the mold between 0~300 mm away from the meniscus is mainly liquid friction stress, and its maximum value is about 145 KPa. Between 300 and 720 mm from the meniscus, solid friction stress is the main stress, and its value increases from 10.6 KPa to 26.6 KPa.

Solid-state friction stress arises due to the failure of the protective slag to melt in time to form a liquid slag layer. During the solidification shrinkage of the casting blank, radial contact pressure is generated between the shell and the mold wall. As the shell moves with the casting drawing, micro-protrusions on the contact surface cause mechanical interlocking. In the presence of a liquid slag film, sliding friction force is formed. Additionally, uneven thermal shrinkage of the shell during solidification shrinkage, coupled with insufficient mold taper, exacerbates local contact pressure, increases friction stress within the mold, causes rupture of the liquid slag film, and forms solid-state friction force.

5. Research on the Control of Friction Stress in Molds

5.1. Influence of Taper on Friction in Molds

The design of the mold taper should not only ensure a close fit between the casting slab and the mold but also prevent solid-state friction between the mold and the slab. Combining the numerical simulation with the calculation results of frictional stress in the mold, it is known that with the progress of casting withdrawal, the frictional stress between the mold and the slab increases, and even the liquid slag film disappears, leading to a severe lubrication failure. Therefore, the mold taper should be reduced. Based on the finite element model and boundary conditions, the mold taper is optimized and reduced to 1.25%·m⁻¹, and the solidification heat transfer process of the slab in the mold is re-numerically simulated using the optimized taper.

(1)

Analysis of temperature field of casting billet after taper optimization

Figure 7 shows the temperature change curve of the corner and the central feature point of the surface from the meniscus to the mold outlet. It can be seen from the figure that near the meniscus, the change trend of the corner is consistent with that of the surface center. However, with the increasing distance from the meniscus, the decline trend of the corner is obviously lower than that of the surface center, and the decline rate is more uniform. At the exit of the mold, the temperature of the corner of the casting blank is 890 °C, the temperature of the surface center is 1045 °C, and the corner temperature is 155 °C lower than the surface center temperature. The temperature of the corner and the center of the surface decreased at the same rate, indicating that the heat transfer rate of the casting billet and the mold did not change much, indicating that the taper was consistent with the solidification shrinkage rate of the medium-carbon special steel, which greatly reduced the formation of mesh cracks on the casting billet surface.

(2)

Air gap analysis between casting billet and mold after taper optimization

Figure 8 shows the air gap change curve of characteristic points on each surface of the casting billet with the distance to the meniscus, which shows the solidification and contraction of each surface of the casting blank in the mold.

It can be seen from Figure 8 that the air gap between the corner of the casting billet and the mold is the largest, followed by the narrow surface, and the smallest gap is at the wide surface. At the distance of 720 mm from the meniscus, the air gap between the center of the corner and the mold is 3.2 mm, the air gap in the center of the wide surface is 1.84 mm, and the air gap in the center of the narrow surface is 2.53 mm. When the distance from the meniscus is 0~200 mm, the growth of the air gap in the corner is more intense, and when the distance is 200~720 mm, the growth trend of the air gap in the corner is slower but more uniform. This is due to the close fit between the casting billet and the mold at the beginning, as faster heat transfer leads to the faster heat loss of the casting billet and the more drastic shrinkage of the shell. With the shrinkage of the billet, the air gap between the mold and the billet increases, so the transverse heat transfer of the billet is reduced, and the contraction of the billet will gradually weaken. The growth rate of the air gap between the center of the wide surface and the center of the narrow surface and the mold is more uniform, which indicates that the mold taper optimization is more reasonable.

5.2. Effect of Physical and Chemical Properties of Casting Powder on Internal Friction Stress of Molds

Good lubrication is one of the key functions of casting powder. The casting powder forms a liquid slag film between the mold and the casting billet, which can effectively reduce the friction between the billet and the mold wall. For medium-carbon steel casting powder, it should have a higher crystallization temperature. An increase in the crystallization rate in the slag film can better control the heat transfer. Meanwhile, the casting powder should also have an appropriate viscosity and a lower melting temperature to ensure lubrication effects [26,27,28].

The main components of the special casting powder of medium-carbon special steel produced by a factory are shown in

Table 7. Main components of casting powder.

Ingredient	SiO2	MgO	CaO	Fe2O3	Al2O3	R2O	F−	C
Content	29.3 ± 5.0	≤5.0	36.5 ± 5.0	≤3.0	5.0 ± 3.0	2.0 ± 1.5	3.8 ± 2.0	10.2 ± 4.0

, and the physical and chemical properties are shown in

Table 8. Physical and chemical properties of casting powder.

Melting Temperature	Melting Speed	Crystallization Temperature	Incubation Time	Inflow Temperature	Viscosity	Transition Temperature
1211 °C	69 s	1262 °C	61 s	1300 °C	0.204 Pa·s	1170 °C

.

At 1300 °C, the viscosity of the casting powder of medium-carbon special steel is 0.204 Pa·s, and the transition temperature is 1170 °C. The comparison between the viscosity temperature curve and the surface temperature of the casting billet is shown in Figure 9a,b.

As can be seen from the Figure 9a,b, near a distance of 300 mm from the meniscus, the cooling rate of the slab surface temperature significantly accelerates, and at this time, the slab surface temperature and the transition temperature of the mold flux are around 1200 °C. At this point, the liquid slag film of the mold flux may disappear completely, leaving only a solid slag film attached to the inner wall of the mold, causing the frictional force between the mold and the slab to transition from a liquid to solid state, leading to a slag film fracture. Therefore, for this medium-carbon special steel, it is necessary to reduce the frictional stress in the mold, lower the transition temperature, prolong the time for the liquid slag film to transform into a solid slag film, and avoid direct contact between the solidified shell and the mold. Previous studies by our research group have found [29] that MgO with a mass fraction of 2% to 4% can reduce the transition temperature of the casting powder and improve the thermal stability. Therefore, by reducing the MgO content in the original slag, increasing the viscosity of the slag to 0.5~0.6 Pa·s, and reducing the transition temperature to about 1100 °C, the friction stress between the casting billet and the mold can be effectively reduced, thus reducing the formation of mesh cracks on the casting billet surface.

If the melting temperature of the casting powder is too high, it cannot form a liquid slag film in time, leading to an increase in solid-state friction between the casting slab and the mold wall, which may cause surface quality defects on the slab. Too slow a melting rate will make the liquid slag layer too thin, resulting in prolonged solid-state friction between the slab and the mold wall. Meanwhile, the frictional force gradually increases over time. Previous studies have found [30] that when the carbon blending mode is changed, the carbon black content is reduced from 1.44% to 0.48% and the graphite content is increased from 1.44% to 2.39%, the particle size of the carbon material is increased, the specific surface area is decreased, and the melting rate is significantly accelerated. In order to reduce the friction in the mold and ensure the appropriate thickness of the liquid slag layer, the carbon mixing mode of the casting powder is changed to increase the carbon black content, so as to reduce the melting temperature of the casting powder to about 1100 °C, and improve the melting speed of the casting powder to the range of 40~50 s, which provides the basis for the liquid slag film to have sufficient lubrication capacity.

Direct contact between the casting slab and the mold causes excessive frictional stress in the mold. The devitrification temperature of the mold flux should be appropriately reduced, and the heat transfer rate on the slab surface should be increased to accelerate its shrinkage rate, avoiding direct contact between the slab and the mold. Studies [31] have found that with the increase in Al₂O₃, the viscosity and melting temperature of the protective slag increase, and the transition temperature decreases. At the same time, the crystallization temperature and crystallization rate decrease with the increase in Al₂O₃. Therefore, the appropriate increase in Al₂O₃ content will reduce the crystallization temperature of the casting powder to about 1150 °C, and improve the surface quality of the casting billet.

The optimization results for the physical and chemical properties of the casting powder of medium-carbon special steel are shown in

Table 9. Optimization results for physical and chemical properties of casting powder.

Viscosity	Transition Temperature	Melting Temperature	Melting Speed	Crystallization Temperature
0.5~0.6 Pa·s	1100 °C	1100 °C	40~50 s	1150 °C

.

5.3. Friction Stress Analysis Between Mold and Billet After Optimization

After adjusting the mold taper and optimizing the physical and chemical properties of the casting powder, the internal friction stress change law of the mold is shown in Figure 10.

As can be seen from Figure 10, after optimization, the friction stress on the surface of the casting billet at 0~550 mm away from the meniscus is dominated by liquid friction, and its value varies in the range of 0~200 Pa. After 550 mm from the meniscus, the thickness of the liquid slag film decreases because the surface temperature of the billet is lower than the transition temperature of the casting powder, and the friction stress on the surface of the billet is mainly solid friction, and its value increases from 20 KPa to 26.6 KPa.

The liquid friction stress is affected by the relative motion state of the mold and the casting blank, and changes with the mold oscillation waveform [5]. As can be seen from Figure 10, at 50 mm away from the meniscus of the mold, the friction stress on the casting billet is relatively large, which is because the casting powder in the early stages of continuous casting is not fully melted and evenly distributed, and there is solid friction between the casting billet and the mold, resulting in greater friction stress on the casting billet. In the range of 50~400 mm from the meniscus, the friction stress in the positive slip-off stage first decreases and then increases. The reason is that in the positive slip-off stage, with the mold oscillation, the oscillation speed first decreases in the same direction as the casting speed, and then increases in the reverse direction with the casting speed. In the range of 400~500 mm from the meniscus, the friction stress in the positive slip-off stage is reduced because the mold oscillation speed is in the same direction as the casting speed. In the range of 50~550 mm from the meniscus, the friction stress in the negative slippage stage. At the same time, all changes in the range of 0~200 Pa, indicating that the friction stress in the negative slippage stage is small and the lubrication is good. At the same time, this also confirms the internal pressure stress in the negative slippage stage, which makes the surface crack heal and is conducive to the release of the billet by the casting mold. Within the range of 500~600 mm from the meniscus, the friction stress in the positive and negative slip-off stages increases sharply because the surface temperature of the casting billet is lower than the transition temperature of the casting powder, and the thickness of the liquid slag film is reduced, making the internal friction in the mold become solid friction, and the friction stress value is about 20 KPa. After adjusting the mold taper and the physical and chemical properties of the casting powder, the frictional force in the mold is significantly reduced compared with before, the lubrication of the liquid slag film is greatly strengthened, and the lubrication performance of the mold is improved, thereby reducing the occurrence of network cracks.

By adjusting the mold taper and optimizing the physical and chemical properties of the mold flux, the frictional stress between the mold and the casting slab is significantly reduced, effectively improving the problem of surface network cracks in medium-carbon special steel. The defect-free medium-carbon special steel produced after adopting the improved process parameters and mold flux properties is shown in Figure 11.

5.4. Relationship Between the Composition of Casting Powder and Friction

At present, the types of casting powders used for different steel grades vary, but the fundamental cause of quality issues such as slab cracks is excessive friction, which may occur between the slab and the mold, or between the slab and the solid casting powder. Previous studies have not clearly established the relationship between the composition of casting powders and friction. Here, we conduct a brief survey of relevant past research, summarizing the most common base materials of casting powders for different steel grades, the relationship between casting powder composition and properties, and the impact of composition on friction.

Table 10. Relationship between composition of casting powder and friction.

Type of Casting Powder	Main Components (Base Materials)	Correlation Between Components and Properties	Relationship Between Components and Friction
Casting powder for high-aluminum steel	CaO, SiO2, Al2O3	Control the content of SiO2 to reduce slag–metal reactions; Li2O lowers the melting temperature, and BaO improves the degree of vitrification, allowing the slag film to uniformly cover the strand shell and reduce cracks caused by local overheating.	Viscosity dominates friction; BaO inhibits crystallization, avoiding friction fluctuations caused by changes in the slag film structure.
Casting powder for high-manganese steel	CaO, SiO2, MnO	MnO inhibits the reaction between Mn and SiO2; Al2O3 stabilizes the melt network structure to avoid abrupt changes in viscosity.	Li2O and F− reduce viscosity to minimize friction; BaO improves spreadability to reduce localized friction concentration.
Casting powder for high-titanium steel	CaO, Al2O3	B2O3 forms a composite network with TiO2 to reduce the precipitation of perovskite and avoid sudden increases in viscosity; CaF2 and B2O3 synergistically reduce the melting point and viscosity, making the thermal resistance of the slag film uniform.	The combination of CaF2 and B2O3 improves fluidity and reduces friction peaks caused by local stagnation.
Casting powder for high-carbon steel	CaO, SiO2	CaO slows down the melting of the casting powder; Al2O3 inhibits the precipitation of cuspidine, avoiding uneven heat transfer caused by the crystalline slag film.	If the carbon content (C) is less than 5%, insufficient slag consumption increases friction; if C exceeds 10%, the casting powder melts slowly, resulting in a thin liquid slag layer and an increase in the friction coefficient.

provides a framework for addressing slab problems caused by friction in the future.

5.5. The Application of This Method in Other Cases

In the series of billet problems solved by this method, typical cases are summarized in

Table 11. The application of this method in other cases.

Steel Grade	Billet Defect	Frictional Stress Before Improvement	Improvement Measure	Frictional Stress After Improvement
Q235	Surface depression	200 KPa	Optimize the viscosity of the casting powder; adjust the secondary cooling water volume.	15~450 Pa
304 stainless steel	Corner crack	120 KPa	Optimize the composition of the casting powder; adjust process parameters.	25~500 Pa
Low-carbon steel	Crack generation in the upper part of the billet shell	9.63~10.75 MPa	Increase the mold vibration amplitude; optimize the casting powder.	10~253 Pa
IF steel	Surface crack	395~620 KPa	Adjust mold parameters; optimize the performance of the casting powder.	50~289 Pa

.

To solve the crack problem caused by excessive friction, the approach starts with reducing the frictional stress between the billet and the mold, as well as between the billet and the protective slag, thereby addressing the surface cracks of billets at the production site. This method is scalable and repeatable, serving as an effective means to solve the issue of excessive friction. By applying this method, we have resolved the surface crack problems of numerous steel grades.

6. Conclusions

Based on the three-layer structure of mold flux within the mold, the frictional stress inside the mold was calculated and analyzed using the relationship between frictional stress, liquid slag film thickness, and viscosity to evaluate the lubrication state between the casting slab and the mold. Using the actual production parameters of 40Mn2 steel and combining with the ANSYS solidification shrinkage simulation results, it was found that the frictional stress within the mold is primarily liquid frictional stress between 0 and 300 mm from the meniscus, with a maximum value of approximately 145 KPa; between 300 and 720 mm from the meniscus, it is primarily solid frictional stress, which increases from 10.6 KPa to 26.6 KPa. The calculation results indicate that excessive frictional stress within the mold, leading to poor lubrication of the casting slab, is the main cause of surface network cracks on the casting slab.

In order to reduce the frictional stress in the mold and improve the lubrication performance of the casting slab, the mold taper was reduced from 1.6%·m⁻¹ to 1.25%·m⁻¹. Meanwhile, the physical and chemical properties of the mold flux were optimized: the viscosity of the mold flux was increased from 0.204 Pa·s to the range of 0.5–0.6 Pa·s, the transition temperature was reduced from 1170 °C to 1100 °C, the melting temperature was decreased from 1211 °C to 1100 °C, the melting rate of the mold flux was increased to the range of 40–50 s, and the devitrification temperature of the mold flux was reduced to about 1150 °C.

After adjusting the mold taper and optimizing the physical and chemical properties of the mold flux, the mathematical analysis shows that within the range of 0–550 mm from the meniscus, the frictional stress in the mold is significantly reduced to approximately 0–200 Pa, and the variation in frictional stress is relatively uniform. The results indicate that the frictional stress in the mold is appropriate, and the casting slab is well-lubricated in the mold. Finally, the problem of surface network cracks in 40Mn2 steel is solved.