ISIJ International
Online ISSN : 1347-5460
Print ISSN : 0915-1559
ISSN-L : 0915-1559
Regular Article
Investigation on Thermo-mechanical Behavior of Mold Corner for Continuous Casting Slab
Du FengmingWang Xudong Liu YuWang ShanjiaoZhang ZeYao Man
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2015 Volume 55 Issue 10 Pages 2150-2157

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Abstract

Corner cracks in slab and billet have always been the important issues during continuous casting. For the purpose of investigating heat transfer, stress and distortion behaviors of corner regions for the slab continuous casting mold, a full-scale stress model of mold which integrated finite-element method with inverse heat transfer algorithm was developed. An inverse algorithm was applied to calculate the heat flux through the measured temperatures from thermocouples buried inside mold plates. Based on this, a full-scale finite-element stress model, including four copper plates, nickel layer and water slots of different depths, was built to reveal the complex thermo-mechanical behavior of mold corners. The results show that the corner regions generate large stress and obvious stress concentration. The mold bends toward the molten steel and it appears V-shaped distortion in the corner region. And the area of water slots decrease after distortion, but it would not apparently affect heat transfer. The cooling effect could be optimized by changing the position, depth and angle of sloped water slots, which helps to flexibly improve the solidification behavior of slab corner.

1. Introduction

The presence of corner cracks in slabs and billets is an important issue during continuous casting, since these cracks will reduce product quality and affect the subsequent rolling process, leading to an increase in grinding and product degradation. The formation of corner cracks is closely related to the conditions under which the corner is cooled improperly, particularly in connection with initial solidification in the mold. The water quantity, the types and depths of water slots, and the stress and distortion of the mold all play important roles in the solidification of the slab corner, and result in the formation and development of corner cracks.

Many mathematical models describing the thermo-mechanical behavior of strands and molds have been proposed in recent years. Samarasekera et al.1) developed a three-dimensional elastic–plastic model, in the belief that the use of a mold with a small radius could reduce corner cracks. O’Conner et al.2) developed an elastic–plastic–creep finite-element model to predict temperature, distortion, and stress in a funnel-shaped mold for thin slabs. Thomas3) predicted the temperature, distortion, and residual stress in a continuous casting mold for slabs using two- and three-dimensional models. Huespe et al.4) proposed a model for the analysis of thermal stress arising during the early stage of a continuous casting process; they combined a finite-element method with a finite-volume method to analyze the stress and strain on a round billet. Janik et al.5) used a finite-element method to investigate the thermo-mechanical behavior of a steel billet in a continuous casting mold. Liu et al.6) developed a three-dimensional finite-element model to predict the stress and distortion of the mold under different conditions, and analyzed in detail the effect of copper plate parameters on the distributions of temperature and distortion. Yin et al.7) determined mold distortion, shell thickness, and gap size by inversely calculating the thermal resistance between the mold and billet. Xie et al.8) focused mainly on thermal stress analysis and conducted experiments to validate the model used. Wang et al.9) investigated the thermal and mechanical behavior of a solidifying shell using a finite-element model. However, the scarcity of reported investigations to explore corner behavior has presented an obstacle to further understanding of the cause and prevention of corner cracks.

In the present work, a two-dimensional full-scale model of mold is developed, taking account of the measured temperature along with design factors such as water slots and nickel layer. The model consists of two parts. The first is an inverse heat transfer model that obtains the real heat flux from temperatures measured by thermocouples embedded in the mold. The second component is a full-scale thermo-mechanical model in which the heat flux obtained from the inverse model is set as a boundary condition. This second component focuses on analysis of heat transfer and on the stress and distortion behavior of the corner regions, allowing the causes of corner cracks to be determined and providing a basis for mold design.

2. Model

2.1. Inverse Heat Transfer Model

Owing to the complexity of the continuous casting process, the mold cannot be assumed to be in an ideal heat transfer state, so it is necessary to obtain the real heat flux between the mold and the slab. In this work, an inverse algorithm is applied to calculate the heat flux using temperatures measured by thermocouples embedded in the mold. In the calculation, the initial values of the heat flux at the measurement points are given, so the mold temperature field can be found, after which the calculated temperatures at the thermocouple locations are compared with the measured temperatures, and the heat flux is adjusted until the calculated and measured temperatures satisfy the convergence condition; the real heat flux is then obtained. Based on this procedure, a two-dimensional inverse heat transfer model is constructed; more information about this model can be found in a previous publication.10)

2.2. Finite-element Stress Model

The temperature fields are input as a thermal load into the stress model, which uses the same mesh as the heat transfer model. The following assumptions are made:

(1) The mold is assumed to be in a plane stress condition.

(2) The mechanical properties of the mold at high temperatures are temperature-dependent.

(3) The effects of creep of copper and nickel are ignored.

(4) The nodes of all hot surfaces are free, while those of all cold surfaces are fixed.

The total strain increment Δ{ε} is composed of an elastic strain increment Δ{ε}e, a plastic strain increment Δ{ε}p, and a thermal strain increment Δ{ε}T:   

Δ {ε} =Δ {ε} e +Δ {ε} p +Δ {ε} T (1)
Here, Δ{ε}T=({α}+d[D]e−1{σ}/dT)dT, where {α} is the coefficient of thermal expansion and [D]e is the elasticity matrix.

In the elastic region, the incremental relationship between stress and strain is   

Δ{σ}= [D] e (Δ{ε}-Δ {ε} T ) (2)
where Δ{σ} is the stress vector; in the plastic region, the relationship is   
Δ {σ} = [D] ep (Δ {ε} -Δ {ε} T )+Δ {σ} T (3)
where   
Δ {σ} T = [D] e (( σ ¯ H)/({σ}T))dT H'+ { σ ¯ /{σ}} T [D] e ( σ ¯ / {σ} ) ,
[D]ep is the elastic–plastic matrix, H is the strength function, and σ is the equivalent stress of the nodes.

All the nodes of the cold surface are fixed, and the boundary conditions of the stress model are   

U x =0,    U y =0 (4)
where Ux and Uy are the displacements in the x and y directions of the nodes on the cold surfaces.

2.3. Calculation Procedure

The calculation consists of two parts, as shown in Fig. 1. The temperature field is first calculated using the real heat flux obtained from the inverse heat transfer model, and then the thermal results are loaded as a boundary condition on the stress model for stress and distortion analysis.

Fig. 1.

Schematic diagram of model.

3. Experimental Details and Calculation Parameters

A finite-element model is developed, taking account of the water slots and the nickel layer, as shown in Fig. 2.10) The mesh and dimensional diagrams of the parts of the mold are shown in Fig. 3.10) The experiment was conducted on a curved caster with a radius of 10.75 m and a metallurgical length of 28.8 m, which cast a slab of thickness 320 mm with width varying from 1800 to 2700 mm at a casting speed of 0.65 m/min. The mold was oscillated in a non-sinusoidal mode with a stroke length varying from 2 to 6 mm and an oscillation frequency varying from 1 to 5 Hz. More details of the caster are given in Table 1 and the main compositions of the steel are shown in Table 2.10) Table 3 gives the related geometrical and process parameters.10) The thermal and thermo-mechanical properties of the materials are given in Tables 410) and 5.

Fig. 2.

Schematic diagram of model.

Fig. 3.

Mesh and dimension diagram.

Table 1. Main design details of caster.
ItemValue or Type
Machine typeCurved, 10.75 m radius
Metallurgical length, m28.8
Mold oscillationNon-sinusoidal
Oscillation stroke, mm2–6
Oscillation frequency, Hz1–5
LubricationPowder
Mold flux feedingManual
Mold level sensorRadioactive sensor
Table 2. Compositions of the steel.
ItemCMnSiPSCrCuSnNi
value, %0.1451.20.250.0180.00360.0320.05780.010.0195
Table 3. Geometry and process parameters.
Itemvalue
Mold Height, mm900
Copper plate thickness, mm40
Slab width, mm2100
Slab thickness, mm320
Water slot width, mm5
Water slot depth, mm
Inside radius:16, 20
Outside radius:12, 16
Narrow face:12, 16
Casting speed, m/min0.65
Meniscus level, mm100
Water temperature, °C
Inlet:31.6
Outlet for inside radius:35.7
Outlet for outside radius:37.3
Outlet for narrow face:37.2
Water flow, L/min
Inside radius:5860
Outside radius:4440
Narrow face:650
Table 4. Thermophysical properties of materials.
MaterialSpecific heat J/(kg·K)Thermal conductivity W/(m·K)Density kg/m3
Copper3903408900
Nickel460808910
Water42000.5998
Table 5. Thermomechanical properties of materials.
MaterialT, °Cα, °C−1T, °CE, GPaσy, MPaEt, GPa
Copper1515.2e-62012833011
7115.7e-620012828011
12716.5e-635012824011
22717.6e-650012816511
32718.3e-6
Nickel20–20016.7e-620–40023070073

*The symbols T, α, E, σy, Et represent the temperature, coefficient of thermal expansion, Young’s modulus, yield strength, and linear hardening slope, respectively.

4. Results and Discussion

4.1. Validation of Model

The calculated and measured temperatures agree remarkably well, and the deviation changes little, as shown in Fig. 4.10) The temperature fluctuates cyclically along the width direction for both the inside and outside radius, which is consistent with the distance between the water slots. The distance between the two deep water slots is 26 mm, while it is 12 mm between the two shallow water slots. The high-temperature peaks are located at the centerline between the two deep water slots as a result of the lack of cooling water at long distances. The average measured temperatures at the outside radius are lower than those at the inside radius by 20°C because the thermocouples at the outside radius are located further from the hot surface. The temperature differences between the temperature peaks and valleys are also different, due to the different distances from the hot surface and the different depths of the water slots. As the edges of the copper plates are not in contact with the molten steel, the temperature rapidly decreases to the cooling water temperature.

Fig. 4.

Comparison of calculated and measured temperatures along the width direction at the positions of the embedded thermocouples: (a) inside radius; (b) outside radius.

According to heat transfer of molten steel, the real heat flux could be obtained by measuring the temperature difference and flow quantity of cooling water. Figure 5 shows the comparison of measured and calculated heat flux of four copper plates. The value of calculated heat flux is very close to that of real heat flux, the deviation varies from 2%–9%, as shown in Fig. 5, which proves the rationality of the model.

Fig. 5.

Comparison of measured and calculated heat flux of four copper plates.

4.2. Temperature Distributions of Corners

Figure 6 shows the sectional temperature distributions at the four corners for different heights.10) The temperature gradient is clearly larger near the hot surface, and the contours become sparse as the distance from the cold surface decreases. At the meniscus, the temperature of the hot surface is approximately 110–150°C and that of the cold surface is about 40–60°C; the temperature difference between the hot and cold surfaces varies from 50°C to 110°C. At the same height, the temperature distributions of the four corners are asymmetric considering of the non-uniform slab shrinkage and slag distribution, and the temperature difference correlates positively with mold temperature. The temperature of the deep water slots is higher than that of the shallow water slots by approximately 10°C in the meniscus, because the deep water slots are closer to the hot surface. Under the two-dimensional heat conduction of cooling water, the temperature of the corner is always lower than the temperatures of the other positions, and the temperature of outer corner decreases to approximately 40°C due to the no direct contact with the molten steel. Owing to the narrow face taper, the temperature of the narrow face is higher than that of the wide face near the corners 100 mm below the meniscus, and the hot surface temperature differences between the narrow face and wide face are approximately 20–30°C. On account of the lack of water slots near the narrow face corner, the sloped off-corner water slots are designed to enforce a cooling effect, and the temperatures of the sloped water slots are almost equal to those of the adjacent deep water slots.

Fig. 6.

Temperature distributions of corner transverse sections at different heights: (a) meniscus; (b) 100 mm below meniscus; (c) 100 mm above bottom.

4.3. Stress Distributions of Mold Corners

Figure 7 shows the sectional stress distributions of the four corners at different heights. Along the casting direction, the range of variation of the stress at the hot surface is approximately 300 MPa, while it is less than 100 MPa at the cold surface. At the meniscus, the root stress of the deep water slot is larger than that of the shallow water slot by 30–80 MPa. The stress is concentrated in the regions of the hot surface, the roots of the water slots, and the corners. Since the temperature there is high and the thermo-mechanical properties of copper and nickel are different, a large stress gradient can be found around the hot surface. The stress distributions and stress values are similar at the inside and outside radius because of the mass distribution of the cooling water, which validates the rationality of mold design. Under the mechanical contact between the wide and narrow faces, large stress values and a significant stress concentration are generated in the corner regions, with the maximum stress at the corners reaching 800 MPa at a height of 100 mm below the meniscus, which exceeds the yield strength of copper plate. These results agree with the calculation results of Samarasekera,11) Zhou,12) Park,13) Duan14) and Cui.15)

Fig. 7.

Stress distributions of corner transverse sections at different heights: (a) meniscus; (b) 100 mm below meniscus; (c) 100 mm above bottom.

In the meniscus region, the stress around the sloped water slot is about 100–200 MPa, which is similar to that at the adjacent deep water slot, and the stress and temperature of the sloped water slot are also close to those of the deep water slot at the other heights, as shown in Figs. 6 and 7. To investigate the vertical distance from the water slot root to the hot surface on the variation of temperature and stress, 6 positions of water slot root in two narrow plates are chosen in Fig. 8. The vertical distance of root A-A’, B-B’ and C-C’ away hot face are 29 mm, 25 mm and 26 mm, respectively. The temperature and stress distributions of the chosen water slot root along the casting direction are shown in Fig. 9. It can be seen that although the design of the sloped water slot is different from that of the deep water slot, their stress and temperature are similar. And for the same copper plate, the stress and temperature of deep water slots root are almost the same, and the stress and temperature of shallow water slots root are also very close, as shown in Fig. 10. Above all, it illustrates that the temperature and stress mainly depend on the vertical distance from the water slot root to the hot surface. Based on this conclusion, the cooling effect could be optimized by changing the position, depth, and angle of the sloped water slots, allowing flexibility in improving heat transfer in the corner region.

Fig. 8.

Schematic diagram of left-up corner and right-up corner.

Fig. 9.

Temperature and stress distributions of water slot root along the casting direction.

Fig. 10.

Temperature (a) and stress (b) distributions of transverse sections of inside radius at different heights.

4.4. Distributions of Stress and Distortion

Six positions, 10, 20, and 30 mm from the corner on the hot surface of the wide and narrow faces, are chosen to analyze the distributions of stress and distortion around the corner along the longitudinal direction, as shown in Fig. 11. Figure 12 shows the stress and distortion distributions of the six positions along the longitudinal direction. From the top of the mold, the stress rapidly increases to a maximum at 70–100 mm below the meniscus, then slowly decreases with weakening heat transfer, and finally slightly increases at the exit because of the presence of the ends of the water slots 25 mm above the mold exit. Among the three positions on the wide face, the stress at position 1 is the largest, with a maximum of approximately 350 MPa, because it is closest to the corner. The stress distributions at the three positions on the narrow face show the same tendency as those on the wide face, with the maximum positions almost the same, although the stress on the narrow face is larger than that on the wide face by approximately 20–50 MPa.

Fig. 11.

Schematic diagram of upper left corner.

Fig. 12.

Stress distributions along the casting direction: (a) wide face; (b) narrow face.

The distortion of the mold shows the same tendency as the stress, also increasing rapidly from the top and reaching a maximum 100 mm below the meniscus, then slowly decreasing until it rises again at the exit. Since the corner is constrained by the mechanical force of the two copper plates, the closer the position is to the corner, the smaller is the distortion. The greatest distortion at position 1 is only 0.05 mm, while that at position 3 is 0.075 mm. Overall, the narrow face exhibits larger distortion, with the maximum distortions at positions 1′ and 3′ being 0.065 and 0.09 mm, respectively.

4.5. Corner and Slot Distortion

Previous studies have shown that the shell thickness of the corner in the lower part of the mold is less than that at the center positions. It is generally believed that two-dimensional heat transfer in the corner region leads to rapid growth of the shell in the meniscus, with a thick air gap appearing as a result of slab shrinkage. Heat transfer then becomes slow, leading to a thin shell in the corner region.

Figure 13 shows a schematic diagram of mold distortion (the distortion value in the figure is magnified 100 times). The mold expands with heat, but the wide faces are connected to the narrow faces through clamping pressure, so the mold bends toward the molten steel under the combination of stress and constraints. It should be noted that V-shaped distortion appears in the corner region, which means that a larger gap will be generated by mold distortion and slab shrinkage. Although the mold distortion is small, with an order of magnitude of only 0.01–0.1 mm, this is so close to that of the air gap, especially in the meniscus region with intense heat transfer, that it will affect the heat transfer of the initial shell, providing another reason for shell growth lag and stress concentration in the slab corner.

Fig. 13.

Schematic diagram of mold distortion at the meniscus.

As can be seen from Fig. 13, the water slots deform concavely, so the flow area of cooling water is changed. In order to investigate whether distortion of the water slots will affect heat transfer, the distortion difference is defined by   

ΔA= A ori - A def (5)
where Aori is the original cross-sectional area of a water slot and Adef is the deformed cross-sectional area. The rate of change of area is determined by   
Δε= A ori - A def A ori (6)
Figure 14 shows the area changes of six water slots, with the slot number being labeled as in Fig. 13. The distortions of the sloped water slots and the deep water slots are larger than those of the shallow water slots, owing to the shorter distance from the hot surface. The area change of the water slots increases with depth. The distortion of deep water slot 5 is relatively large, with the area change being about 0.00427 mm2 and the rate of change of area being 0.0039 mm2. In general, the area of a water slot decreases after distortion, but the rate of change of area is relatively small, within 0.004%, so this subtle change in area will not have an obvious effect on heat transfer, and the effect on water quantity can be ignored.
Fig. 14.

Area changes of water slots.

5. Conclusions

A full-scale stress model of the mold, integrating a finite-element method with an inverse heat transfer algorithm, has been developed to investigate heat transfer, stress, and distortion in the corner region of a continuous casting mold.

(1) The stress is concentrated in the regions of the hot surface, the water slot roots, and the corners. Large stress values and a significant stress concentration are generated in the corner regions, with the maximum stress at the corners reaching 800 MPa at 100 mm below the meniscus. Along the casting direction, the range of variation of the stress on the hot surface is approximately 300 MPa, while it is below 100 MPa at the cold surface.

(2) Along the casting direction, the stress rapidly increases to a maximum at 70–100 mm below the meniscus, with a maximum value of approximately 300–400 MPa, then slowly decreases with weakening heat transfer, and then slightly increases at the exit. The distortion of the mold shares the same tendency as the stress, with the maximum distortion varying from 0.065 to 0.09 mm.

(3) The stress and temperature of a sloped water slot are close to those of a deep water slot. The temperature and stress mainly depend on the vertical distance from the water slot root to the hot surface. The cooling effect could therefore be optimized by changing the position, depth, and angle of the sloped water slots, which allows some flexibility in improving the solidification behavior at the slab corner.

(4) The mold bends toward the molten steel, and a V-shaped distortion appears in the corner region. The order of magnitude of the distortion is close to that of the air gap, leading to a larger gap in the high-flux region, providing another reason for shell growth lag and stress concentration at the slab corner.

(5) The area of a water slot decreases after distortion, with the change in area increasing with the depth of the slot, although the rate of change of area is relatively small, within 0.004%, so this change in area should not have an obvious effect on heat transfer.

Acknowledgments

We would like to acknowledge the financial support of the National Natural Science Foundation of China (51474047/51004012). This project was granted financial support from the China Postdoctoral Science Foundation (2012M520621 / 2013T60511). Furthermore, the support of the High Technology Research and Development Program of China (2009AA04Z134) is gratefully acknowledged. And it is also supported by the Fundamental Research Funds for the Central Universities.

References
 
© 2015 by The Iron and Steel Institute of Japan
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