Generation Mechanism of Unsteady Bulging in Continuous Casting-2 -FEM Simulation for Generation Mechanism of Unsteady Bulging-

Abstract

In the continuous casting of steel, unsteady bulging contributes to degradation of slab quality. It has been reported that unsteady bulging is promoted by uneven solidification in the mold, but the effect of uneven solidification on unsteady bulging has not been clarified. In this study, a Finite Element Method (FEM) simulation was conducted. Shell deformation was calculated by an elasto-plastic analysis assuming that the slab moves between the rolls, considering time dependency. The bulging value and mold level fluctuation, which change corresponding to the solidified shell thickness, ferrostatic pressure and roll pitch, were obtained.

In the simulation results, the shell is deformed by ferrostatic pressure. The bulging shell pushes out under the rolls in the thickness direction, and unsteady bulging occurs. While the shell is passing through rolls with the same pitch, unsteady bulging becomes larger. When the solidified shell is uneven, stress concentrates on the thinner portions, and this stress concentration accelerates unsteady bulging even at the same average shell thickness. Based on these results, an operational index for suppressing unsteady bulging by reducing unevenness of the solidified shell was proposed.

1. Introduction

In the continuous casting of steel, mold level fluctuations contribute to degradation of slab quality. In particular, the phenomenon in which the mold level fluctuates at the roll pitch period is called mold level fluctuation caused by unsteady bulging, and is a problem that must be solved in order to realize quality assurance and stable operation. The cause of unsteady bulging has been studied by simulations in the past.^1,2) Matsumiya et al.³⁾ created a mathematical analysis model of slab bulging based on bending-shear theory for a beam. Nakajima et al.⁴⁾ calculated the bulging behavior by a dynamic model in which the solidified shell is likened to a beam moving on rolls. The reported calculated bulging amount was in good agreement with measured values. Kanai et al.⁵⁾ obtained the occurrence limits of unsteady bulging and the cause of mold level fluctuation due to unsteady bulging by a two-dimensional beam model.

Furthermore, various approaches have been taken to suppress mold level fluctuation caused by unsteady bulging. Kitada et al.^6,7) reported that unsteady bulging occurs at the same intervals as the movement of depletion between rolls, and cycle variation for unsteady bulging was suppressed by changing the mold level position in casting. Kato et al.⁸⁾ increased the solidified shell thickness at a particular location by increasing the amount of secondary cooling water, thereby reducing the mold level fluctuation caused by unsteady bulging. Yoon et al.⁹⁾ have reported that unsteady bulging was suppressed by adoption of a non-cyclic roll pitch.

Yamagami et al.¹⁰⁾ have reported that mold level fluctuation by unsteady bulging is fostered by uneven solidification in the mold. They proposed a mechanism by which uneven solidification in the mold is caused by a peritectic reaction, and bulging then increases and mold level fluctuation caused by unsteady bulging occurs. In addition, uneven solidification is fostered by increased mold level fluctuation.

Nevertheless, the effect of uneven solidification on unsteady bulging has not been quantitatively determined in previous studies. Therefore, in this study, we determined the bulging and mold level fluctuation values by FEM simulation, and investigated the cause of unsteady bulging and the mechanism by which uneven solidification is promoted by uneven solidification.

2. Experimental Method

2.1. Heat Transfer Analysis Model

A two-dimensional (2-D) heat transfer analysis model was used, as shown in Fig. 1. The slab size of the model was 1/2 of the thickness of a practical slab and the L cross section of the unit width at the center of slab width, and assumed that no heat conduction occurs in the width direction. By dividing the lattice L cross section, the temperature of the lattice points was obtained by discretizing by the differential method in a 2-D heat conduction equation considering time dependence, as shown in Eq. (1).   

$$C\rho \frac{\partial T}{\partial t}=\frac{\partial }{\partial x}\left(k\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left(k\frac{\partial T}{\partial y}\right)$$(1)

C: specific heat (kJ/kg/K), ρ: density(kg/m³), k: thermal conductivity(W/m/K), T: temperature (K)

Fig. 1.

Heat transfer analysis model.

Teshima et al.¹¹⁾ reported that the cooling state of the slab surface is considered to comprise cooling by water spray, cooling by mist spray, cooling by convection and cooling by radiation, and estimated the average heat transfer coefficient in the secondary cooling zone.

In the present study, the cooling condition of the L cross section on the slab surface was given as the average heat transfer coefficient. The slab surface was given the distribution of the heat transfer coefficient in the casting direction and reproduced the unevenness of the solidified shell thickness.

2.2. Unsteady Bulging Analysis Model

Figure 2 shows the elasto-plastic analysis model of unsteady bulging in the FEM simulation. σ_x and ε_x are stress and strain in the casting direction, and σ_y and ε_y are stress and strain in the thickness direction. The slab size and mesh size were set to be the same as in the heat transfer analysis. The roll was assumed to be rigid and fixed at the axial center. The temperature distribution of the slab was given by the result of the 2-D heat transfer analysis, and deformation resistance was given according to the temperature at each lattice point. Table 1 shows the material property values used in the simulations. Young’s modulus, Poisson’s ratio and deformation resistance were the actually measured values as a function of temperature in low carbon steel. The exterior value was used for the physical properties at a temperature higher than 1150°C.

Fig. 2.

Elasto-plastic analysis model.

Table 1. Material property of carbon steel in this study.

Temperature (°C)	Young’s modulus (N/mm2)	Poisson’s ratio (–)	Deformation resistance (N/mm2)
550	1.69E+05	0.320	2.92E+02
700	1.48E+05	0.335	1.62E+02
850	1.29E+05	0.350	4.84E+01
1000	1.22E+05	0.355	3.15E+01
1150	1.01E+05	0.370	1.74E+01
1300	7.80E+04	0.385	7.32E+00
1480	2.08E+04	0.403	3.66E+00

When the temperature was higher than the solidus temperature, the phase was treated as a liquid portion. The deformation resistance of the liquid portion was 0 [N/m²]. The ferrostatic pressure at the boundary between the solid and liquid phases was given considering the bulging force of the slab. As shown Fig. 2, since displacement was constrained in the head portion of a slab which was cut to a length of from 1 m to 2 m, the slab was assumed not to be deformed in the casting or thickness directions.

The deformation behavior of the slab was obtained by elasto-plastic analysis when the slab moved between rolls. The bulging value and mold level fluctuation, which change corresponding to the solidified shell thickness, ferrostatic pressure and roll pitch, were also obtained. Figure 3 shows a schematic diagram of the concept of fluctuation of the liquid portion.

Fig. 3.

Schematic drawing of displacement of liquid in elasto-plastic analysis.

In the analysis of unsteady bulging, the slab bulged between the rolls as a result of ferrostatic pressure, then the solidified shell was pushed back in the thickness direction by the rolls. This means that the liquid portions, which have lower rigidity, were pushed out in the upstream direction of the segment.

The displacement of liquid portion at x is defined as U_x. The average displacement at the end surface of C cross–section (x=0) was defined as U_ave. When the amplitude of U_ave becomes steady under each condition, the maximum value of U_ave was evaluated as the maximum amount of displacement (U_max) of the liquid portion while a slab having a length of from 1 m to 2 m moves between the rolls in the casting direction.

3. Results

3.1. Results of Heat Transfer Analysis

Figure 4 shows the temperature profile of the slab surface obtained by 2-D heat transfer analysis. Figure 4(a) shows an example of the temperature distribution of the uniform solidified shell, and Fig. 4(b) shows that of the uneven solidified shell. When the heat transfer coefficient was constant at 800 W/m²/K in the casting direction, the solidified shell thickness was uniform. On the other hand, when a distribution of the heat transfer coefficient in the casting direction was given, a difference in the slab temperature existed in the casting direction and the solidified shell became uneven. Heat transfer coefficients of 1000 W/m²/K and 600 W/m²/K were given at pitches from 300 to 50 mm in the casting direction. When the average of the solidified shell was equal in the uniform and uneven solidified shells, the average temperature of the slab surface was also equal.

Fig. 4.

Morphology of solidified shell calculated by 2-D heat transfer analysis.

Unevenness of the solidified shell was evaluated by the index U_d,¹²⁾ as shown in Eq. (2), where D₁ is the minimum shell thickness, D₂ is the maximum shell thickness, D_ave is the average shell thickness and L is the uneven length of the solidified shell in the casting direction, respectively.   

$${\mathrm{U}}_{\mathrm{d}}=\left({\mathrm{D}}_2-{\mathrm{D}}_1\right)/\mathrm{L}$$(2)

An elasto-plastic analysis was performed by using the temperature distribution obtained by the heat transfer analysis as the initial condition.

3.2. Bulging Analysis of Uniform Solidified Shell

The bulging amount of the slab is influenced by the roll pitch and ferrostatic pressure, which are specifications of the continuous casting machine, in addition to the solidified shell thickness, which is determined by the cooling conditions and casting speed. First, employing the uniform solidified shell thickness obtained by the heat transfer analysis as an initial condition, an elasto-plastic analysis was performed for different conditions of the average thickness of the solidified shell, roll pitch and ferrostatic pressure, as shown in Table 2. Figure 5 shows the relationship between the solidified shell thickness and displacement of the liquid in the casting direction when the roll pitch and ferrostatic pressure were constant. The solidified shell thickness was analyzed under seven thickness conditions from 50 mm to 80 mm, assuming a constant thickness in the casting direction (uniform solidified shell thickness). Only in the simulation of the shell thickness of 50 mm, the slab length in the casting direction was analyzed for the two conditions of 1 m and 2 m, and when the slab length was 2 m, the displacement of the liquid was calculated as being half of that under the 1 m condition in order to evaluate the mold level fluctuation per slab unit length. For the other shell thicknesses, U_max was evaluated only for the slab length of 1 m.

Table 2. Uunsteady bulging analysis condition at uniform solidified shell.

	Roll pitch (mm)	Ferrostatic pressure (MPa)	Shell thickness (mm)
Case-1	280	1.00	50–80
Case-2	250–350	1.00	50
Case-3	280	0.25–1.50	50

Fig. 5.

Relationship of solidification shell thickness and displacement of liquid in casting direction (Case 1).

The bulging amount increased as the solidified shell thickness decreased. The displacement of the liquid also increased under this condition. The displacement of the liquid was almost equal at the slab lengths of 1 m and 2 m. However, irrespective of the slab length, the displacement of the liquid per slab unit length was constant in the steady state in this model. This result supports the appropriateness of the assumption that the displacement of the head portion of the slab was constrained and the casting direction and thickness direction do not deform, as shown Fig. 2.

Figure 6 shows the relationship between the roll pitch and displacement of the liquid in the casting direction when the solidified shell thickness and ferrostatic pressure were constant and the slab length was 1 m. When the roll pitch was increased from 250 mm to 350 mm, the bulging amount increased with the roll pitch, followed by an increase in the displacement of the liquid.

Fig. 6.

Relationship of roll pitch and displacement of liquid (Case 2).

Figure 7 shows the relationship between the ferrostatic pressure and displacement of the liquid in the casting direction under the condition of the same shell thickness and roll pitch. When the ferrostatic pressure was increased from 0.25[MPa] to 1.5[MPa], the bulging amount of the slab increased with increasing ferrostatic pressure, and again, displacement of the liquid also increased.

Fig. 7.

Relationship of ferrostatic pressure and displacement of liquid (Case 3).

3.3. Bulging Analysis of Uneven Solidified Shell

An elasto-plastic analysis was performed under the unevenly solidified shell conditions obtained by the heat transfer analysis. Table 3 shows the elasto-plastic analysis conditions for the uneven solidified shell. The roll pitch and ferrostatic pressure were set to be constant.

Table 3. Unsteady bulging analysis condition at uneven solidified shell.

		Roll pitch (mm)	Ferrostatic pressure (MPa)	Shell thickness (mm)			Ununiform pitch: L (mm)
				Dave	D1	D2
Case-4	(a)	280	1.00	52.5	50	55	50
	(b)			58.5	57	60
	(c)			68.5	67	70

In Case 4(a), the average solidified shell thickness was 52.5 mm, the difference of the minimum and maximum solidified shell thickness was 5 mm and the uneven pitch was 50 mm. In Case 4(b) and Case 4(c), the average solidified shell thicknesses were 58.5 mm and 68.5 mm, respectively. In both cases, the difference of the shell thickness was 3 mm and the uneven pitch was 50 mm. The slab length was constant at 1 m.

Figure 8 shows the relationship between the solidified shell thickness and displacement of the liquid in the casting direction, comparing the cases when the solidified shell was uniform and uneven. In the uneven solidified shell, the bulging amount increased as the solidified shell thickness decreased, and displacement of the liquid also increased. This was the same behavior as that of the uniform solidified shell. However, with the uneven solidified shell, the bulging amount increased more than with the uniform solidified shell, and displacement of the liquid was larger than in the case of the uniform solidified shell, even though the average shell thickness was equal.

Fig. 8.

Relationship of solidification shell thickness and displacement of liquid in casting direction (Case 4).

4. Discussion

4.1. Generation Mechanism of Unsteady Bulging

Figure 9 shows an example of the profile of the bulging amount in the thickness direction of the points on the slab surface when the slab moved to a segment in the downstream direction. Equation (3) is the calculation formula for the maximum deflection value by the elastic beam model proposed by Kawawa et al.¹³⁾   

$$\delta =60/384\times \mathrm{P}{\mathrm{L}}^4/\mathrm{E}{\mathrm{d}}^3$$(3)

δ (cm): maximum deflection value, P (kg/cm²): ferrostatic pressure, L: roll pitch (cm), E (kg/cm²): Young’s modulus, d (cm): solidified shell thickness

Fig. 9.

Bulging amount of slab surface (shell thickness: 50 mm).

In addition, Komoda¹⁴⁾ proposed the deflection value of an elastic beam, B. H. Knell¹⁵⁾ proposed the deflection value of an elastic flat plate and Morita et al.¹⁶⁾ proposed the bulging amount by stress analysis, as shown respectively by Eqs. (4), (5) and (6).   

$$\delta =12/384\times \mathrm{P}{\mathrm{l}}^4/\mathrm{E}{\mathrm{d}}^3$$(4)

  

$$\delta =\left(1/16\right)\times \left(\mathrm{P}{\mathrm{l}}^2/\mathrm{E}{\mathrm{d}}^3\right)\times \left\{\left(\mathrm{l}2/2\right)-3\mathrm{\nu }{\mathrm{d}}^2+\mathrm{\nu }{\left(\mathrm{b}+\mathrm{d}\right)}^2\right\}$$(5)

  

$$\delta =\mathrm{P}/220\times \mathrm{l}3.5/{\mathrm{d}}^3$$(6)

ν (–): Poisson’s ratio, b (cm): slab width

Using the above equations, the maximum bulging amount was calculated under the conditions in this study. As a result, the bulging amount was from 0.12 to 1.2 mm. It is considered that these values correspond to the steady bulging value when the casting length was from 0 m to 1.5 m, as shown Fig. 9. According to the simulation, the maximum bulging amount was 0.47 mm in this area, and this value is consistent with the range of the maximum bulging amount in past findings.

As the generation mechanism of unsteady bulging, it is considered that the plastically deformed solidified shell does not remain in its original shape, and the shell is pushed back by the rolls, as Yoon et al.⁹⁾ reported.

In this simulation, as shown in Fig. 9, the bulging amount increased with movement of the slab, and negative deformation occurred at the roll pass line. This phenomenon means the deformed shell is pushed back in the thickness direction by the rolls.

As shown in Figs. 10(a) and 10(b), the solidified shell bulges between the rolls due to ferrostatic pressure, the bulging shell climbs over the next roll and the bulging shape remains even after the next roll because of plastic deformation. The cause of plastic deformation of the shell is considered to be that the strength of the solidified shell is insufficient because of the increased ferrostatic pressure, increased roll pitch and decrease of shell thickness due to the changing casting speed and secondary cooling conditions. The plastically deformed shell is pushed back directly below the roll, and the liquid steel is pushed out Δd in the upstream direction. It is considered that this upstream flow appears as a mold level fluctuation H. As shown Fig. 10(c), in a section having the same roll pitch, the bulging portion is pushed out all at once.

Fig. 10.

Schematic diagram of occurrence of unsteady bulging.

Figure 11 shows the relationship between the displacement of the liquid and the distance of the unsteady bulging occurrence position. In the early state after unsteady bulging occurs, mold level fluctuation increases gradually. The mold level fluctuation becomes steady when the occurrence position moves downstream to some degree. From the above results, it is estimated that unsteady bulging continues to expand because the roll pitch is the same.

Fig. 11.

Relationship of distance of casting direction and displacement of liquid (Case 1).

4.2. Effect of Uneven Solidification on Unsteady Bulging

In the simulation of the uniform solidified shell thickness in Case 1, strain of the solidified shell increased as the shell thickness became thinner, and since plastic deformation became more likely to occur, displacement of the liquid increased. In Case 2 and Case 3, the bulging amount between the rolls increased with increased roll pitch or with increased ferrostatic pressure. In these cases, the amount of molten steel pushed out at the roll increased, and this resulted in an increase in the displacement of the liquid. With the uneven solidified shell thickness, Yamagami et al.¹⁰⁾ reported that unsteady bulging is increased by uneven solidification in the mold.

In this study, with the uneven solidified shell, the displacement of the liquid increased in comparison with the displacement in case of the uniform solidified shell. Figure 12(a) shows the distribution of the maximum principal stress in the uniform solidified shell (Case 1), and Fig. 12(b) shows the distribution of the maximum principal stress in the uneven solidified shell (Case 4(a)) in this simulation.

Fig. 12.

Distribution of maximum principal stress in simulation.

In Case 4, in which the displacement of the liquid increased with the uneven solidified shell, stress concentrates on the uneven portion of the shell. As a result, deformation of the shell increases, and the bulging amount and displacement of the liquid also increase. Thus, this simulation revealed that unsteady bulging is fostered by the unevenness of the solidified shell, even at the same shell thickness.

The moment of inertia I (mm⁴), which represents the amount of deformation resistance for the bending moment, is expressed as Eq. (7).   

$$\mathrm{I}=\mathrm{b}{\mathrm{d}}^3/12$$(7)

The theoretical value δ of deflection of the slab cross section C considering the moment of inertia is given by Eq. (8).   

$$\delta =\mathrm{w}{\mathrm{l}}^4/\left(384\mathrm{E}\mathrm{I}\right)$$(8)

w (kgf): load, l (mm): length

Figure 13 shows the value of the moment of inertia for the distance in casting direction in Case 4(a). The moment of inertia becomes minimum at the position where the thickness of the shell is at its minimum. The theoretical value δ for deflection of the slab cross section C becomes maximum at the portion of minimum shell thickness.

Fig. 13.

Moment of inertia in casting direction.

The distribution of the maximum principal stress in the simulation also increased at the minimum shell thickness point. Deformation of the shell is fostered at the uneven portion (D₁) of the shell. It is considered that the bulging amount and displacement of the liquid also increase.

5. Conclusion

As the result of a FEM simulation of the bulging amount and mold level fluctuation in continuous casting, the following results were obtained.

(1) The simulation successfully reproduced the mold level fluctuation due to unsteady bulging. The solidified shell bulged between the rolls due to ferrostatic pressure, the plastically deformed shell was then pushed back at the roll and the liquid steel moved upstream in the segment.

(2) In the early stage after the occurrence of unsteady bulging, mold level fluctuation increased gradually. Then, the fluctuation became steady after the slab moved downstream to some extent. Based on the simulation, it is estimated that unsteady bulging continued to expand because the roll pitch remains the same.

(3) Unsteady bulging is worsened by the unevenness of the solidified shell, even when the average shell thickness is the same. In the case of uneven solidification, stress concentrates on the uneven portion of the shell and deformation of the shell becomes worse. As a result, the bulging amount and displacement of the liquid also increase.

The simulation in this study makes it possible to determine the range of operational indexes considering uneven solidification.

References

• 1)   K.  Okamura and  H.  Kawashima: Tetsu-to-Hagané, 75 (1989), 1905.
• 2)  J-D. Lee and C-H. Yim: ISIJ Int., 40 (2000), 765.
• 3)   T.  Matsumiya,  H.  Kajioka and  Y.  Nakamura: Tetsu-to-Hagané, 68 (1982), A145.
• 4)   K.  Nakajima,  A.  Kanazawa,  Y.  Sugiya,  J. Y.  Lamant and  M.  Kawasaki: Tetsu-to-Hagané, 71 (1985), S1058.
• 5)   N.  Kanai,  S.  Horioka,  T.  Okada and  H.  Saiki: Trans. Jpn. Soc. Mech. Eng. C, 65 (1999), No. 631, 1207.
• 6)   H.  Kitada,  M.  Kawamoto and  T.  Murakami: CAMP-ISIJ, 17 (2004), 854.
• 7)   H.  Kitada,  T.  Murakami and  M.  Kawamoto: CAMP-ISIJ, 22 (2009), 1037.
• 8)   T.  Kato,  K.  Ishii,  S.  Kohira,  J.  Kubota and  K.  Tsutsumi: CAMP-ISIJ, 17 (2004), 853.
• 9)   U.-S.  Yoon,  I.-W.  Bang,  J. H.  Rhee,  S. Y.  Kim,  J.-D.  Lee and  K. H.  Oh: ISIJ Int., 42 (2002), 1103.
• 10)   A.  Yamagami,  C.  Matsumura,  H.  Yamamoto,  N.  Tsudome,  T.  Sera and  A.  Yoshii: Tetsu-to-Hagané, 73 (1987), S913.
• 11)   T.  Teshima,  T.  Kitagawa,  S.  Miyahara,  H.  Funanokawa,  K.  Ozawa and  K.  Okimoto: Tetsu-to-Hagané, 74 (1988), 1282.
• 12)   H.  Murakami,  M.  Suzuki,  T.  Kitagawa and  S.  Miyahara: Tetsu-to-Hagané, 78 (1992), 105.
• 13)   T.  Kawawa,  H.  Sato,  S.  Miyahara,  T.  Koyano and  H.  Nemoto: Tetsu-to-Hagané, 60 (1974), 486.
• 14)   K.  Komoda: 27th and 28th Nishiyama Memorial Seminar, ISIJ, Tokyo, (1970).
• 15)   B. H.  Knell: Steel Times, 18 (1967), 189.
• 16)   Y.  Morita,  H.  Kawashima and  M.  Nakamura: Sumitomo Met., 35 (1983), 271.