Prediction Model for Degree of Solid-shell Unevenness during Initial Solidification in the Mold

Abstract

The unevenness of the solid-shell and heat flux (Φ_q) during solidification were measured for various steel grades with casting apparatus. Then the data were used to develop a model to predict the degree of solid-shell unevenness according to the steel composition. The surface of the solid-shell was the least uniform when the steel composition was near the peritectic point, at which Φ_q was the lowest. A model to calculate an index of solid-shell unevenness was developed under the following conditions. Solid-shell unevenness is formed at the initial solidification under the constant cooling rate, which occurs only where solid fraction (fs) is from 0.9 to 1.0. In addition, solid-shell unevenness is proportional to the amount of phase transformation and inversely proportional to the temperature range of ΔT_PT; between the temperature of fs = 0.9 and the solidus temperature. This model shows about 87% of the prediction accuracy in the experimental results of this study and also gives reasonable explanations of solid-shell unevenness observed in previous researches on steels.

1. Introduction

When a solid-shell of steel grows non-uniformly in the mold, the strain increases,¹⁾ and which can easily cause longitudinal cracking of cast steel. In addition, if the solid-shell is uneven, heat transfer is disturbed by air gaps that form between the solid-shell and the mold wall; this disturbance extends the time for which the initial solid-shell is exposed to high temperature, and thereby promotes coarsening of grains on the surface of the cast steel.^{2,3,4,5,6,7,8)} These coarse grains cause concentration of segregation and stress at grain boundaries, so surface cracking is facilitated.^9,10,11,12) During continuous casting, peritectic steels in which the solid-shell is unevenly solidified also have relatively large mold level fluctuations, which degrade the surface quality of the cast steel.^13,14,15) One way to minimize the surface defects on cast steel produced by continuous casting is to design chemical compositions that yield an evenly-solidified solid-shell. Furthermore, casting conditions such as mold flux, mold oscillation, and tundish superheat should be chosen carefully by considering the degree of solid-shell unevenness.^16,17)

The composition of the steel affects its initial solidification behavior and heat flux (Φ_q) in the mold. Φ_q in the specific carbon region is low due to air gaps that form as a result of peritectic transformation.^18,19,20) However, the solid-shell unevenness may not precisely coincide with Φ_q of the mold,²¹⁾ because both phenomena are also affected by other factors including casting speed^21,22) superheat,^4,23) mold lubricant,^3,18,24,25) oscillation conditions and mold-level fluctuation.^3,21,26,27) Some researchers have attempted to quantify the unevenness of solid-shell by measuring the solidification profile of steel according to the carbon concentration ([C]). Many studies showed that the degree of solid-shell unevenness was large at [C] ≈ 0.1 wt%,^{28,29,30,31,32)} but some studies showed large solid-shell unevenness even at ultra-low [C].^33,34)

During solidification, peritectic steel grades undergo relatively large volume shrinkage.^35,36,37) As a result, the unevenness of solid-shell due to peritectic transformation can be severe during continuous casting, so longitudinal cracking can easily occur. Longitudinal cracking depends upon the extent of peritectic transformation, so the tendency of steels to form delta-ferrite (δ) during solidification is an important factor³⁸⁾ and the tendency can be expressed as a ferrite potential. Studies of ferrite potential have attempted to predict the severity of longitudinal cracking by calculating the degree to which the composition of steel is separated from the peritectic point,^38,39,40) and relationships presented in these studies have been widely used to predict the occurrence of longitudinal cracks during continuous casting; i.e., ferrite potential has been used as a useful indicator to predict the possibility of longitudinal cracks.

The goal of this study is to develop a model that can predict the degree of solid-shell unevenness according to steel composition. When developing a new steel grade, this model can be used to design a composition that can minimize the non-uniformity of the solid-shell or effectively optimize the casting operations considering the non-uniformity of the solid-shell.

2. Experiments

Thirteen steels (Table 1) were tested. All samples were machined to cylinders (diameter 43 mm, 950 g) for easy charging into an alumina crucible. A vacuum/pressure casting machine (VTC 200 V, Indutherm) was used; it consisted of a melting chamber and a casting chamber (Fig. 1(a)). The melting chamber fully melts the specimen, then rotates and pours the molten steel into the copper (Cu) mold of the casting chamber. The melting chamber has an induction coil that can heat the specimen in an alumina crucible; a pyrometer is installed to measure the temperature of the molten steel in the crucible. During melting and casting processes, the chambers were evacuated to ≤ 10⁻³ Torr, then purged with Ar gas to prevent oxidation of the specimen. The superheat of the molten steel was set to 50°C. Inside the Cu mold in the casting chamber, cooling water was supplied to the lower part and passed through a water channel to the upper part. The cooling water circulation device was used to control the temperature and flow rate of cooling water. Two thermocouples were installed at different depths, one thermocouple (TC#1) was 5 mm deep from the Cu mold surface, and the other (TC#2) was 10 mm deep from it as shown in Fig. 1(c). The thermocouples inside the Cu mold are connected to a data logger, which records temperature changes in 0.1-s intervals, then calculates the Φ_q by using the Fourier heat-conduction equation   

$$\begin{array}{l}Heat\mathrm{\hspace{0.17em} }Flux\mathrm{\hspace{0.17em} }(MW/m^2)\\ =\mathrm{\Delta }{T}_{TC\#1-TC\#2}*thermal\mathrm{\hspace{0.17em} }conductivity\mathrm{\hspace{0.17em} }of\mathrm{\hspace{0.17em} }Cu/0.005\mathrm{\hspace{0.17em} }m\end{array}$$(1)

where ΔT_TC(TC#1−TC#2) is the temperature difference between two thermocouples inserted at different depths at the same position; i.e., 5 mm apart. Surfaces of specimens in contacted with the Cu mold were observed under an optical microscope to get the surface 3D-profile. The 3D-profile was then converted to seven 2D profiles by its vertically cutting with 2.5-mm intervals. Five positions were observed for each specimen. Maximum depths of 2D profiles were measured (Fig. 1(b)) and the profile depth was the average of the maximum depths. The measured profile depth was used as the degree of solid-shell unevenness.

Table 1. Chemical compositions of test specimens (wt.%).

Code	wt.%									liquid fraction at the start of peritectic reaction
	C	Si	Mn	P	S	Ni	Cu	Ti	Nb
Steel 1	0.03	–	0.02	0.006	0.008	0.01	0.02	–	–	< 0.00
Steel 2	0.09	0.21	0.35	0.015	0.015	0.05	0.16	–	–	0.01
Steel 3	0.09	0.22	0.37	0.015	0.016	0.06	0.17	–	–	0.02
Steel 4	0.06	0.26	1.44	0.006	0.001	0.45	0.14	0.01	0.01	0.04
Steel 5	0.05	0.01	1.90	0.007	0.002	0.90	0.28	0.01	0.01	0.05
Steel 6	0.06	0.28	1.92	0.006	0.002	0.50	0.18	0.01	0.03	0.08
Steel 7	0.14	0.60	0.55	–	–	0.20	0.10	–	–	0.15
Steel 8	0.15	0.28	1.31	0.011	0.004	0.01	0.01	–	0.02	0.23
Steel 9	0.19	0.25	0.50	–	–	–	–	–	–	0.26
Steel 10	0.28	0.22	0.36	0.015	0.012	0.06	0.18	–	–	0.48
Steel 11	0.37	0.21	0.35	0.014	0.011	0.01	0.02	–	–	0.70
Steel 12	0.44	0.23	0.60	0.010	0.002	0.01	0.01	–	–	0.95
Steel 13	0.56	0.24	0.67	0.013	0.011	0.01	0.02	–	–	> 1.00

Fig. 1.

Schematic diagram of experimental equipment.

3. Result and Discussion

3.1. Heat Flux and Surface Profile of Test Specimens

The meaured heat flux (Φ_q) of each specimen was shown in Fig. 2. Specimens were listed in order of liquid fraction at the start of peritectic reaction (f_lsp). This choice was made because a rapid volume contraction occurs due to phase transformation during solidifcation, and the liquid fraction during the phase transformation can affect the solid-shell unevenness. The f_lsp of each specimen was calculated using a commercial software (JmatPro). Φ_q of specimens with a carbon of 0.28% or more was greater than those of specimens with a carbon contents of less than 0.2%. It was the smallest in specimen 6 with an f_lsp of 0.08.

Fig. 2.

Heat flux of test specimens. Top x-axis: liquid fraction at the start of peritectic reaction for each specimen.

Each specimen’s surface roughness of specimen was shown as 3D profle (Fig. 3(a)) and profile depth (Fig. 3(b)). When the initial liquid fraction at the start of the peritectic (f_lsp) was 0.08 or less, the profile depth increased with increase of f_lsp. On the other hand, in case that f_lsp was larger than 0.08, the profile depth decreased with increase of f_lsp. The largest profile depth was 1125 μm in specimen 6. The profile depth became very small when the f_lsp exceeded 0.48. The profile depth and Φ_q were inversely proportional to each other (Fig. 4), and the correlation coefficient value was 0.82 for all specimens, but it was 0.93 except for specimens with f_lsp of 0.48 or more as shown in Fig. 4.

Fig. 3.

Surface profile of test specimens; (a) Surface 3D profile, (b) profile depth.

Fig. 4.

Relation between the heat flux and the profile depth for (a) all test specimens and (b) test specimens except high carbon specimens (steel 10–13).

Figure 5 shows the behavior of profile depth and Φ_q according to f_lsp. When f_lsp was less than about 0.4, the profile depth and Φ_q were inversely proportional to each other, but when f_lsp was greater than about 0.4, the profile depth decreased as Φ_q decreased. According to the f_lsp, it can be thougth of as follows to have an opposite relationship between two values; profile depth and Φ_q. When the f_lsp was approximately 0.4 or more, the profile depth became about 200 μm or less and the effect of solid-shell unevenness on the heat flux became relatively small. Also, the temperature of molten steel decreased as the carbon concentration increased due to applying the same superheat for each steel. Eventually the heat flux decreased owing to the lower temperature of molten steel.

Fig. 5.

Correlation between profile depth and heat flux according to liquid fraction at the start of peritectic reaction. (Online version in color.)

3.2. Uneven Solid-shell Index

The formation of solid-shell unevenness can be thought in a viewpoint of solidification shrinkage.

In the initial solidification, molten steel contacts the mold and solidification proceeds. At this stage, the contact state between the molten steel and the mold is very good regardless of the steel composition. Therefore, it can be regarded that the cooling rate of molten steel is constant for all steel composition during initial solidification.

As the temperature decreases, the liquid phase transforms into a solid phase and the volume contracts. If the liquid phase moves to the area where the volume contraction occurs to compensate for the volume contraction, deformation of the solid phase is difficult to occur. Therefore, it is considered that only the volume contraction that occurs while the liquid phase is difficult to move can cause the deformation of the solid phase. Figure 6(a)⁴¹⁾ is a schematic diagram showing the correlation between the flow of the residual liquid phase and the solid fraction (fs). It was known that the liquid phase could not be moved or penetrated in the area of the solid fraction above 0.9,^37,42) which means the solid fraction at the liquid impenetrable temperature (LIT). It can be said that the solid-shell unevenness occurs in the range of fs from 0.9 to 1.0 (ΔT_PT). The ΔT_PT according to the carbon concentration was shown in Fig. 6(b).

Fig. 6.

Schematic drawings for temperature range where solid shell unevenness forms. a) Relation between the temperature range of solid shell unevenness formation and dendritic structure, b) Calculation temperature range (T) for solid-shell unevenness according to carbon content, LIT ≤ T ≤ T_fS = 1.0, ⓐ is the L/δ transformation, ⓑ is the (L + δ)/γ transformation, ⓒ is the L/γ transformation. (Online version in color.)

The chemical composition of a steel affects its phase-transformation speed, and distortion occurs severely when the phase transformation speed is fast.⁴³⁾ In other words, as the transformation speed increases, the amount of deformation causing distortion increases. Besides, at a constant cooling rate, the phase transformation speed increases as the ΔT_TP decreases (Fig. 6(b)). When solidification is complete, that is, below the solidus temperature, the solid shell is sufficiently strong and it is difficult to deform the solid shell due to subsequent phase transformation.

Based on the above review results, the model is suggested under the following three conditions.

1) During initial solidification, the contact between the molten steel and the mold is perfect. Therefore, the cooling rate of molten steel is constant regardless of the steel composition.

2) The solid phase deformation that causes solid-shell unevenness occurs only at temperatures between LIT and T_{fS = 1.0} (ΔT_PT).

3) The amount of deformation increases as phase transformation speed increases. Therefore, it is inversely proportional to the ΔT_PT and proportional to the amount of phase transformation.

The degree of deformation occurrence, that is, the degree of solid-shell unevenness, was defined as Uneven Solid-shell Index (USI) (Eq. (2)).   

$$\mathrm{U}\mathrm{S}\mathrm{I}=\text{a}\times \frac{\mathrm{\Delta }{f}_{(L/\delta )}}{\mathrm{\Delta }{T}_{(L/\delta )}^{\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }n}}+\mathrm{b}\times \frac{\mathrm{\Delta }{f}_{(L+\delta )/\gamma }}{\mathrm{\Delta }{T}_{(L+\delta )/\gamma }^{\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }n}}+\mathrm{c}\times \frac{\mathrm{\Delta }{f}_{(L/\gamma )}}{\mathrm{\Delta }{T}_{(L/\gamma )}^{\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }n}}$$(2)

Where Δf_(L/δ) is the amount of liquid transformed to delta ferrite (Fig. 6(b), area ⓐ), and Δf_(L+δ)/γ is the amount of peritectic transformation (Fig. 6(b), area ⓑ), Δf_(L/γ) is the amount of liquid transformed to austenite (Fig. 6(b), area ⓒ), and ΔT_(i/j) is the temperature range at which the phase is transformed from i to j.

a, b and c are average volume changes according to the phase transformation for 13 steel grades.

Each value was a = 0.0028 b = 0.0033 and c = 0.0040.

Changes of phase fraction (Δf_(L/δ), Δf_(L+δ)/γ and Δf_(L/γ)) and ΔT_PT (ΔT_(L/δ), ΔT_(L+δ)/γ and ΔT_(L/γ)) (Fig. 7) were calculated using JMatPro (version 11). The consistency between the profile depth and the calculated USI value in Eq. (2) was investigated using the f_LIT and an exponent term of ΔT_PTⁿ as variables. The consistency between the two factors was the best when f_LIT = 0.9 and n = 0.3. The USI showed a good match with the measured profile depths (R² = 0.87; Fig. 7(a)). In addition, it was confirmed that the composition region with a large profile depth could be accurately predicted by USI (Fig. 7(b)).

Fig. 7.

Comparison of USI calculation result and profile depth. a) Relation between USI and Profile depth, b) Behavior of USI and profile depth according to liquid fraction at the start of peritectic reaction.

This result confirmed that the USI gave a good explanation of the solid-shell unevenness caused by phase transformation during initial solidification of the steel in the mold.

Figure 8 shows an example of the calculation results for two steel grades and the yellow part indicates fs from 0.9 to 1.0. The calculated results of the phase fraction change for steel 6 were Δf_(L/δ) = 2.4, Δf_(L+δ)/γ = 49, and each transformation temperature range was ΔT_(L/δ) = 2.72°C and ΔT_(L+δ)/γ = 2.67°C (Fig. 8(a)). The calculation result of steel 8 was shown in Fig. 8(b). Each phase fraction change and transformation temperature range were Δf_(L+δ)/γ = 17.4, Δf_(L/γ) = 7.2, and ΔT_(L+δ)/γ = 0.76°C, ΔT_(L/γ) = 10.6°C respectively. Because of these differences, in particular the differences in Δf_(L+δ)/γ and ΔT_(L+δ)/γ, the USI changed according to the steel composition.

Fig. 8.

Example of USI calculation for specimens. a) steel 6, b) steel 8, the calculation temperature range is from T_fs = 0.9 to T_fs = 1.0 (Yellow box).

The developed prediction model was compared with the measured solid-shell unevenness of the previously reported results (Fig. 9). The components of the three steel grades compared were Si 0.2, Mn 0.5, P 0.02, S 0.02, Al 0.5 wt% (Fig. 9(a)),²⁹⁾ Si 0.3, Mn 0.3, P 0.01, S 0.01, Al 0.03 wt% (Fig. 9(b))²⁸⁾ and Si 0.21–0.25, Mn 1.17–1.25, P 0.011–0.015, S 0.001–0.002, Cr 0.12–0.179, Al 0.035–0.049 wt% (Fig. 9(c)).^31,32) As shown in Fig. 9, the calculated USI values showed excellent consistency with previously studied results.

Fig. 9.

Relation between USI and measured results in previous studies of solid-shell unevenness.

These results confirmed that the USI of this paper could accurately predict the degree of solid-shell unevenness according to the steel composition. We believe the USI can be useful to guide the design of chemical compositions of new steel grades, or for optimization of casting operation to achieve the stable surface quality of slabs. In the future, this model will be developed to predict the solid-shell unevenness of high-alloy steels such as high Mn steel, high Si/Ni steel and stainless steel.

4. Conclusion

To quantify the effect of the steel composition on solid-shell unevenness, the profile depth and Φ_q were measured for 13 steel grades. A model to predict the degree of solid-shell unevenness was developed according to the steel composition. The main results are as follows.

(1) An apparatus and method that uses a 950 g ingot were developed to accurately measure solid-shell unevenness and heat flux according to the steel composition.

(2) The model to predict solid-shell unevenness was developed under the following conditions:

(a) The molten steel has very good contact with the mold at initial solidification. Therefore, the mold cooling rate is constant regardless of the steel composition.

(b) Solid-shell unevenness occurs between T_{fS = 0.9} and T_{fS = 1.0} (ΔT_PT).

(c) The solid-shell unevenness is proportional to the amount of phase transformation (Δf) and inversely proportional to the temperature range of the phase transformation (ΔT_PT).

The USI is   

$$0.0028\times \frac{\mathrm{\Delta }{f}_{(L/\delta )}}{\mathrm{\Delta }{T}_{(L/\delta )}^{\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }0.3}}+0.0033\times \frac{\mathrm{\Delta }{f}_{(L+\delta )/\gamma }}{\mathrm{\Delta }{T}_{(L+\delta )/\gamma }^{\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }0.3}}+0.0040\times \frac{\mathrm{\Delta }{f}_{(L/\gamma )}}{\mathrm{\Delta }{T}_{(L/\gamma )}^{\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }0.3}}$$

(3) The developed model (USI) in this study showed approximately 87% consistency with the experimental results, and could also explain the previously reported solid-shell unevenness behaviors.

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