**3.1. Finite element model**

560 Numerical Simulation – From Theory to Industry

where Ex is the equivalent elastic modular ratio, expressed as:(Sheng Y et al ,1993)

The stress of bending and straightening is expressed as:(Lei H et al ,2007)

*s i i*

> Casting speed (m·s-1)

**Table 2.** Comparison of slab stresses in the bending zone( FU JianXun et al.2011(a))

Casting speed (m·s-1)

tensile stress acting on the contact surface.

Slab shell thickness (10-2 m)

Slab shell thickness (10-2 m)

Roller ID

Roller ID

The stress of disalignment is expressed as: (Chen J,1990)

*<sup>D</sup> <sup>S</sup>*

<sup>x</sup> 10 , <sup>100</sup> *So m so T T <sup>E</sup> T*

6

1

*s s <sup>i</sup>*

2 2

1 1 100%, <sup>2</sup>

*D D Ri R*

<sup>300</sup> 100%, *is*

The values of these stresses calculated for the Q235 slab are listed in Tables 2 and 3. The calculations were based on the parameters of the continuous casters of Maanshan Iron and Steel Co. Ltd. A casting speed of 0.0167 m·s-1 was used. The bending zone of the continuous caster is at the 10th~15th rollers of the 2nd segments, and the straightening zone is at the 60th~65th rollers of the 9th segments. A negative stress indicates that a pushing stress acts on the contact surface between the slab and rollers whereas a positive stress indicates a

> Bulging (10-2 m)

10 4.10 0.0167 0.101 0.115 0.0128 -0.0011 11 4.28 0.0167 0.097 0.115 0.0134 -0.0011 12 4.45 0.0167 0.093 0.115 0.0139 -0.0012 13 4.62 0.0167 0.090 0.115 0.0144 -0.0013 14 4.78 0.0167 0.087 0.115 0.0149 -0.0011 15 4.93 0.0167 0.084 0.115 0.0154 -0.0010

> Bulging (10-2 m)

60 10.4 0.0167 0.14 0.196 0.0160 0.0160 61 10.5 0.0167 0.09 0.152 0.0197 0.0197 62 10.6 0.0167 0.08 0.149 0.0198 0.0198 63 10.7 0.0167 0.08 0.147 0.0200 0.0200

 

q 2

*l*

(4)

(6)

Bulging strain(%)

Bulging strain(%) Disalignment strain(%)

Disalignment strain(%)

Bending strain(%)

Bending strain (%)

(5)

Building a satisfactory three-dimensional (3D) finite element model for the numerical simulation of continuous casting in the secondary cooling zone is quite complex. Thus, to simplify the problem, the following assumptions are made, as in our previous work((FU JianXun et al. 2010(b-c); 2011(b)):


Based on these assumptions, the thermal-mechanical coupled model of the whole secondary cooling zone is divided into 6 independent sub-models for calculation. The 15 segments of the secondary cooling zone are divided into 6 groups. The first 5 groups each contain 2 segments; the remaining 5 segments make up the last group as a completely solidified slab. A 2-m slab is used for the simulation. The slab goes through the roll arrangement at a given speed. The simulation is performed continuously from the first group to the last group, and the results of a group of rollers are taken as the initial inputs for the subsequent group.

Eight-node isoparametric elements are used for the geometric discretization of the computational domain in the model. The slab comprises 4500 elements and 5250 nodes. Figure 2 shows the finite element models of the rollers and the slab in the caster. Figure 3(a) shows the rollers and the slab in the 3rd independent sub-model. Figure 3(b) shows the 6th independent sub-model.

Due to the symmetry of the slab in the width direction, one half of the slab was simulated. The grid units at the start plane of the slab move forward at a given speed. The static

pressure of molten steel is taken as a mechanical boundary condition. The boundary is applied to the solidifying front of the slab, which is defined as the position with zerostrength temperature (ZST). Considering the effects of solidification-induced segregation and solid fraction (*f*s), the temperature of the units is the ZST where *f*s is equal to 0.8, and the units are considered a solidified shell where *f*s 0.8. T80 denotes the temperature at the boundary between the solid phase and the liquid phase (*T*80=ZST). Static pressure acts on the units where the temperature is higher than T80. The boundary conditions of heat transfer and contact are also applied to the model.

Numerical Simulation of Slab Broadening in Continuous Casting of Steel 563

(7)

(8)

(9)

. *e ie T ij ij ij ij*

 

is composed of time-independent inelastic strain and time-

is the stress; and *T* is the temperature.

*m*

 

dependent creep deformation. A viscoelastic-plastic model is used to describe the solidifying behavior of the slab under the conditions of continuous casting, which is

> ( , , ). *ie ie ij ij ij*

This equation indicates that the non-elastic strain ratio is a function of the stress,

The constitutive equation of time-dependent plastic deformation is used to describe the stress of carbon steel under various temperatures and strain ratios; it is expressed as: (S.

1/ Aexp( Q / R )[sinh(<sup>β</sup> )] ,. ( ),

, *m* are constants.

1

(10)

When carbon steel becomes plastic, strain hardening is observed. The coefficient of strain

<sup>p</sup> () . *<sup>n</sup> H Kn* 

In this work, the user program includes the strain hardening coefficient in the elastic-plastic model of the MSC. Marc solver to describe the viscoelastic-plastic behavior of the cast slab under high temperature. The work hardening of carbon steel is described by the equations

*T K*

 

 *f T* 

P

  *n*

*P*

 

hardening can be obtained from the following equation:

given by Sorimachi and Brimacombe (K. Sorimachi et al.,1977);

*K*

where:

is the total strain;

is the elastic strain;

The non-elastic strain *ie*

expressed as: (Chen J,1990)

Kobayashi et al.,1988)

*R* is the gas constant;

is the equivalent stress;

*K* is the strength factor;

temperature, and non-elastic strain.

is the equivalent plastic strain ratio;

is the equivalent plastic strain;

*n* is the factor of hardening; and *A* ,

*Q* is the activation energy of deformation;

is the non-elastic strain;

is the thermal strain, and *ij* is the strain tensor.

*ij* 

is the non-elastic strain ratio; *ij*

*ij* 

*e j* 

*ie ij* 

*T ij* 

where: *ie ij* 

where:

*P*

*P* 

**Figure 2.** Finite element model of all the rollers and the slab (FU JianXun et al. 2011(b))

**Figure 3.** Finite element models of (a) the third group of rollers and the slab (b)the sixth group of rollers( FU JianXun et al. 2010(c))

### **3.2. Constitutive equations**

The key factors that determine the accuracy of a model for analyzing the stress in a slab are included in the constitutive equation of the slab. These factors are heat transfer, mechanical load, stress relaxation, and plastic strain, all of which are time-dependent. The constitutive equation of steel at high temperature, which determines the accuracy of numeric simulations, is expressed as:

Numerical Simulation of Slab Broadening in Continuous Casting of Steel 563

$$
\varepsilon\_{\rm ij} = \varepsilon\_{\rm ij}^e + \varepsilon\_{\rm ij}^{ie} + \varepsilon\_{\rm ij}^T. \tag{7}
$$

where:

562 Numerical Simulation – From Theory to Industry

and contact are also applied to the model.

rollers( FU JianXun et al. 2010(c))

**3.2. Constitutive equations** 

simulations, is expressed as:

pressure of molten steel is taken as a mechanical boundary condition. The boundary is applied to the solidifying front of the slab, which is defined as the position with zerostrength temperature (ZST). Considering the effects of solidification-induced segregation and solid fraction (*f*s), the temperature of the units is the ZST where *f*s is equal to 0.8, and the units are considered a solidified shell where *f*s 0.8. T80 denotes the temperature at the boundary between the solid phase and the liquid phase (*T*80=ZST). Static pressure acts on the units where the temperature is higher than T80. The boundary conditions of heat transfer

**Figure 2.** Finite element model of all the rollers and the slab (FU JianXun et al. 2011(b))

 **Figure 3.** Finite element models of (a) the third group of rollers and the slab (b)the sixth group of

The key factors that determine the accuracy of a model for analyzing the stress in a slab are included in the constitutive equation of the slab. These factors are heat transfer, mechanical load, stress relaxation, and plastic strain, all of which are time-dependent. The constitutive equation of steel at high temperature, which determines the accuracy of numeric

*ij* is the total strain; *e j* is the elastic strain; *ie ij* is the non-elastic strain; *T ij* is the thermal strain, and *ij* is the strain tensor.

The non-elastic strain *ie ij* is composed of time-independent inelastic strain and timedependent creep deformation. A viscoelastic-plastic model is used to describe the solidifying behavior of the slab under the conditions of continuous casting, which is expressed as: (Chen J,1990)

$$
\dot{\varepsilon}\_{\text{ij}}^{\text{it}} = f(\sigma\_{\text{ij}}, T, \varepsilon\_{\text{ij}}^{\text{it}}). \tag{8}
$$

where: *ie ij* is the non-elastic strain ratio; *ij* is the stress; and *T* is the temperature.

This equation indicates that the non-elastic strain ratio is a function of the stress, temperature, and non-elastic strain.

The constitutive equation of time-dependent plastic deformation is used to describe the stress of carbon steel under various temperatures and strain ratios; it is expressed as: (S. Kobayashi et al.,1988)

$$\begin{cases} \dot{\overline{\boldsymbol{\varepsilon}}}\_{\mathcal{p}} = \mathcal{A} \exp(-\mathcal{Q}/\mathcal{R}T) [\sinh(\beta \mathcal{K})]^{1/m} \text{.} \\ \overline{\boldsymbol{\sigma}} = \mathcal{K} \cdot (\overline{\boldsymbol{\varepsilon}}\_{\mathcal{p}})^{\mathsf{n}} . \end{cases} \text{.} \tag{9}$$

where:

*P* is the equivalent plastic strain ratio;

*R* is the gas constant;

*Q* is the activation energy of deformation;

is the equivalent stress;

*P* is the equivalent plastic strain;

*K* is the strength factor;

*n* is the factor of hardening; and *A* , , *m* are constants.

When carbon steel becomes plastic, strain hardening is observed. The coefficient of strain hardening can be obtained from the following equation:

$$H = K \cdot n \cdot (\overline{\varepsilon}\_{\mathbb{P}})^{n-1}. \tag{10}$$

In this work, the user program includes the strain hardening coefficient in the elastic-plastic model of the MSC. Marc solver to describe the viscoelastic-plastic behavior of the cast slab under high temperature. The work hardening of carbon steel is described by the equations given by Sorimachi and Brimacombe (K. Sorimachi et al.,1977);

$$K\_1 / E = 0.13 \exp(-0.023 \theta) \,\text{.}\tag{11}$$

Numerical Simulation of Slab Broadening in Continuous Casting of Steel 565

,

*ZST S*

is the Poisson's ratio at

*TT f f* (17)

(16)

(14)

Steel C (%) S i(%) Mn(%) P(%) S(%) Al(%) Tl(°C) Ts(°C) SPHC 0.05 0.05 0.20 ≦0.02 ≦0.012 0.03 1528.9 1493.0 Q235 0.18 0.20 0.40 ≦0.025 ≦0.022 1517.0 1446.0

The coefficient of thermal expansion, Young's modulus of elasticity, and Poisson's ratio of the steel as functions of temperature are required for simulation. The elastic modulus of carbon steel for various temperatures during continuous casting is given in equations (14)

, ,

(968 2.33 (1.9 10 )\*T (5.18 10 ) \* ) \* 10 900,

When *E*=*E*(Ts) and *E*zst=*E*(Tzst), Ezst takes a small non-zero value in order to restrain the deviatoric stress in the liquid phase region to maintain hydrostatic pressure. When the temperature is lower than *T*s, the elastic modulus is expressed by (14); when it is higher than

When the temperature is lower than *T*s, Poisson's ratio can be defined as equation (14); when the temperature is higher than *T*s, with decreasing *f*s, Poisson's ratio gradually increases from the value at Ts to a certain value which is close to 0.5; it remains at this value

<sup>L</sup> ( ) (1 ) , T T 1 , <sup>1</sup>

<sup>80</sup> , , ( ). *ZST S ZST*

ZST; it is very close to 0.5. The Poisson's ratio of the steel for various temperatures is shown in Figure 4(a). The coefficient of thermal expansion values of Q235 and SPHC are taken from

**4. Effects of casting speed on slab broadening (Jian-Xun Fu et al , 2011b)** 

By tracing one node of the slab at the side-face and recording its width, the width of the slab at various positions in the secondary cooling zone can be obtained. The RUB can then be derived from the simulated width of the slab. The simulated RUB values of Q235 and SPHC

*ff f f f <sup>f</sup>*

 


<sup>80</sup> ( 1), *S ZST S T T Tf f* (15)

9

*E T T Ts T*

(347.6525-0.350305\*T) \* 10 900

( ) (1 ) , <sup>1</sup> *S ZST S S ZST ZST*

above ZST, as expressed by equations (16) and (17). (Uehara M et al.,1986)

*s ZST s s ZST*

Where *f*ZST is the solid phase ratio at ZST, often taken as 0.80. *ZST*

 

the local measurement results shown in Figure 4(b).

*ZST*

steels at three casting speeds are shown in Figure 5(a) and (b), respectively.

*E T*

*f f E fE <sup>E</sup> f*

**Table 5.** Compositions of SPHC steel and Q235 steel

and (15).( Ueshima Y et al; 1986, I.Ohnaka,1986)

*T*s, it is expressed by (15).

**4.1. Numerical simulation** 

$$K\_2 \, / E = \begin{cases} 0.045 - 3.87 \times 10^{-5} \,\theta \,\, / \,\theta < 1050^{\circ}C \\ 0.385 \cdot \exp(-0.00422 \,\theta) \,\, / \,\, \theta \ge 1050^{\circ}C \end{cases} \tag{12}$$

$$K\_3 \, / \, E = \begin{cases} 0.0197 - 1.68 \times 10^{-5} \theta \, \text{ } / \, \theta < 1050^{\circ} \text{C} \\ 0.0226 \exp(-0.00223 \theta) \, \text{ } / \, \theta \ge 1050^{\circ} \text{C} \end{cases} \tag{13}$$

where: *E* is the elastic modulus; *K*1~*K*3 are factors of hardening; *Θ* is the temperature (°C).

Equations (11)~(13) are applied when the strain is smaller than 0.01~0.02, and greater than 0.02, respectively. The factors of hardening are incorporated into the elastic-plastic model in the software package Marc by the user programs.

#### **3.3. Parameters**

All simulation parameters are taken from the technological parameters of the #2 continuous caster (SMS-Demag) of Maanshan Iron and Steel Co. Ltd. The parameters of the continuous caster are listed in Table 4.


**Table 4.** Parameters of the slab and caster at various segments

Casting temperature: *T*=1533 °C; Liquidus temperature *T*l=1513 °C; Solidus temperature *T*s =1446.0 °C; *T*80 = 1459.6 °C; Environmental temperature: 25 °C; Roller temperature: 100 °C; Coefficient of contact heat transfer: 25.0 W/(m·K) ( Y. S. Xi and H. H. Chen,2001) ; Coefficient of fraction: 0.3; Distance tolerance: 0.01 ( Y. S. Xi and H. H. Chen,2001).

When the casting speed is in the range of 1.0~1.2 m/min, the SMS-Demag casting machine uses a fixed cooling water intensity in the secondary cooling zone. The parameters of the SPHC steel and Q235 steel are listed in Table 5.


**Table 5.** Compositions of SPHC steel and Q235 steel

2

3

the software package Marc by the user programs.

**Table 4.** Parameters of the slab and caster at various segments

**3.3. Parameters** 

Segment No.

caster are listed in Table 4.

Shrinkage between rollers (mm)

Casting temperature: *T*=1533 °C; Liquidus temperature *T*l=1513 °C; Solidus temperature *T*s =1446.0 °C;

Environmental temperature: 25 °C;

SPHC steel and Q235 steel are listed in Table 5.

Roller temperature: 100 °C;

*T*80 = 1459.6 °C;

<sup>1</sup> *K E*/ 0.13exp( 0.023 ),

 

 

where: *E* is the elastic modulus; *K*1~*K*3 are factors of hardening; *Θ* is the temperature (°C).

Equations (11)~(13) are applied when the strain is smaller than 0.01~0.02, and greater than 0.02, respectively. The factors of hardening are incorporated into the elastic-plastic model in

All simulation parameters are taken from the technological parameters of the #2 continuous caster (SMS-Demag) of Maanshan Iron and Steel Co. Ltd. The parameters of the continuous

1-2 0.20/0.46 0.2375 0.200 0.240 0-4.374 3-4 0.46/0.46 0.2370 0.245 0.284 4.374-8.388 5-6 0.46/0.44 0.2362 0.255 0.297 8.388-12.592 7-8 0.44/0.44 0.2354 0.265 0.310 12.592-16.992 9-10 0/0.30 0.2346 0.283 0.322 16.992-21.254 11-15 0.30/0.30 0.2343 0.300 0.335 21.254-33.249

Coefficient of contact heat transfer: 25.0 W/(m·K) ( Y. S. Xi and H. H. Chen,2001) ; Coefficient of fraction: 0.3; Distance tolerance: 0.01 ( Y. S. Xi and H. H. Chen,2001).

When the casting speed is in the range of 1.0~1.2 m/min, the SMS-Demag casting machine uses a fixed cooling water intensity in the secondary cooling zone. The parameters of the

Roller diameter(m)

Slab thickness (m)

0.045 3.87 10 , 1050 / , 0.385 exp( 0.00422 ) , 1050 *<sup>C</sup> K E*

0.0197 1.68 10 , 1050 / , 0.0226exp( 0.00223 ) , 1050 *<sup>C</sup> K E*

 

 

> 

 

 

5

 

5

(11)

(12)

(13)

Distance from meniscus (m)

*C*

*C*

Slit width between rollers (m)

The coefficient of thermal expansion, Young's modulus of elasticity, and Poisson's ratio of the steel as functions of temperature are required for simulation. The elastic modulus of carbon steel for various temperatures during continuous casting is given in equations (14) and (15).( Ueshima Y et al; 1986, I.Ohnaka,1986)

$$\begin{cases} E = \text{(347.6525-0.350305°T)} \ast 10^9 & , \quad T < 900 \text{ }, \\ E = \text{(968-2.33\*T + (1.9\*10^3)°T^2 - (5.18\*10^{-7})°T^3)^\* 10^9 & , \quad Ts > T \ge 900 \text{ }, \end{cases} \tag{14}$$

$$E = \frac{(f\_S - f\_{ZST}) \cdot E\_S + (1 - f\_S) \cdot E\_{ZST}}{1 - f\_{ZST}}, \quad Tso > T > T\_S (f\_{ZST} \le f\_S \le 1), \tag{15}$$

When *E*=*E*(Ts) and *E*zst=*E*(Tzst), Ezst takes a small non-zero value in order to restrain the deviatoric stress in the liquid phase region to maintain hydrostatic pressure. When the temperature is lower than *T*s, the elastic modulus is expressed by (14); when it is higher than *T*s, it is expressed by (15).

When the temperature is lower than *T*s, Poisson's ratio can be defined as equation (14); when the temperature is higher than *T*s, with decreasing *f*s, Poisson's ratio gradually increases from the value at Ts to a certain value which is close to 0.5; it remains at this value above ZST, as expressed by equations (16) and (17). (Uehara M et al.,1986)

$$\nu = \frac{(f\_s - f\_{ZST}) \cdot \nu\_s + (1 - f\_s) \cdot \nu\_{ZST}}{1 - f\_{ZST}}, \text{ T} > \text{T}\_{\text{L}} \left( f\_{ZST} \le f\_S \le 1 \right), \tag{16}$$

$$\nu = \nu\_{\rm ZST} \quad \text{ } T \ge T\_{80} \quad \text{ } \quad \{f\_{\rm S} \ge f\_{\rm ZST}\}. \tag{17}$$

Where *f*ZST is the solid phase ratio at ZST, often taken as 0.80. *ZST* is the Poisson's ratio at ZST; it is very close to 0.5. The Poisson's ratio of the steel for various temperatures is shown in Figure 4(a). The coefficient of thermal expansion values of Q235 and SPHC are taken from the local measurement results shown in Figure 4(b).

#### **4. Effects of casting speed on slab broadening (Jian-Xun Fu et al , 2011b)**

#### **4.1. Numerical simulation**

By tracing one node of the slab at the side-face and recording its width, the width of the slab at various positions in the secondary cooling zone can be obtained. The RUB can then be derived from the simulated width of the slab. The simulated RUB values of Q235 and SPHC steels at three casting speeds are shown in Figure 5(a) and (b), respectively.

Numerical Simulation of Slab Broadening in Continuous Casting of Steel 567

values at the exit of the caster were 0.76%, 0.96%, and 1.14%, respectively. For SPHC steel, when the casting speeds were 1.0, 1.1, and 1.2 m/min, the maximum RUB values were 1.34%, 1.44%, and 1.69%, respectively, and the RUB values at the exit of the caster were

Slab broadening is closely correlated with casting speed, which may be due to the slab's temperature changing with casting speed. When the casting speed increased, the liquid core length and temperature of the slab both increased. With increasing temperature of the slab, the high-temperature mechanical properties of the slab changed; ductility increased and the

With increasing casting speed, a given cross section of the slab takes up the same amount of space. At 25 m away from the meniscus, the surface temperature in the wide face at a casting speed of 1.2m/min is 11.6 °C and 21.7 °C higher on average than those at casting speeds of 1.1 and 1.0 m/min, respectively (see Figure 6). Under a given set of conditions, increasing the casting speed increases production. However, high casting speed can lead to

**0.0 0.2 0.4 0.6 0.8 1.0**

**Distence from center, m**

For continuous caster #2 at Maanshan Iron and Steel Co. Ltd., the slab widths of two steel grades were tracked online at the exit of the caster (the end of the 15th segment); the slab width was measured once per minute. For each grade of steel, measurements were taken for

Figure 7(a) shows the slab width and the RUB of SPHC steel at various moments. Slab broadening can be clearly seen. The RUB of SPHC steel ranges from 1.4% to 2.4%, with an average of 1.96%. The average RUB is greater than the ratio of linear shrinkage, indicating that the width of the slab after cooling was greater than the top width of the mold. This

 c**asting speed 1.0m/min casting speed 1.2m/min casting speed 1.1m/min**

**Figure 6.** Surface temperature on the slab at various positions.(FU JianXun et al. 2011(b))

more than 70 minutes. The data are shown in Figures 7(a), and (b), respectively.

result shows that slab broadening occurred in the secondary cooling zone.

strength and resistance to external pressure decreased, increasing the RUB.

0.64%, 0.76%, and 0.95%, respectively.

**900**

**4.2. Verification of simulation results** 

**920**

**940**

**960**

**Temperature, °C**

**980**

**1000**

slab broadening.

**Figure 4.** (a) Young's modulus of elasticity and Poisson's ratio of Q235 steel for various temperatures; (b)Coefficient of thermal expansion of Q235 and SPHC. (Jian-Xun Fu et al , 2011b)

**Figure 5.** (a) RUB values versus distance from meniscus of a Q235 steel;.(b) RUB values versus distance from meniscus of a SPHC steel at three casting speeds. (FU JianXun et al. 2011(b))

Slab broadening for Q235 and SPHC steels at three casting speeds shows similar characteristics. The values of the RUB at the three casting speeds are all positive in the whole secondary cooling zone, which means that slab broadening existed for Q235 and SPHC steels at these speeds. The RUB changed from one segment to another for the first five segments. The RUB increased and then gradually decreased after reaching its maximum at the fifth and sixth segments. Near the tenth segment, the RUB decreased smoothly; the slab became completely solidified at this location.

The simulations of Q235 and SPHC steels produced similar results. The RUB increased with increasing casting speed. For Q235 steel, when the casting speeds were 1.0, 1.1, and 1.2 m/min, the maximum RUB values were 1.44%, 1.88%, and 2.04 %, respectively, and the RUB values at the exit of the caster were 0.76%, 0.96%, and 1.14%, respectively. For SPHC steel, when the casting speeds were 1.0, 1.1, and 1.2 m/min, the maximum RUB values were 1.34%, 1.44%, and 1.69%, respectively, and the RUB values at the exit of the caster were 0.64%, 0.76%, and 0.95%, respectively.

Slab broadening is closely correlated with casting speed, which may be due to the slab's temperature changing with casting speed. When the casting speed increased, the liquid core length and temperature of the slab both increased. With increasing temperature of the slab, the high-temperature mechanical properties of the slab changed; ductility increased and the strength and resistance to external pressure decreased, increasing the RUB.

With increasing casting speed, a given cross section of the slab takes up the same amount of space. At 25 m away from the meniscus, the surface temperature in the wide face at a casting speed of 1.2m/min is 11.6 °C and 21.7 °C higher on average than those at casting speeds of 1.1 and 1.0 m/min, respectively (see Figure 6). Under a given set of conditions, increasing the casting speed increases production. However, high casting speed can lead to slab broadening.

**Figure 6.** Surface temperature on the slab at various positions.(FU JianXun et al. 2011(b))

## **4.2. Verification of simulation results**

566 Numerical Simulation – From Theory to Industry

**Figure 4.** (a) Young's modulus of elasticity and Poisson's ratio of Q235 steel for various temperatures;

(a) (b)

**Figure 5.** (a) RUB values versus distance from meniscus of a Q235 steel;.(b) RUB values versus distance

(a) (b)

Slab broadening for Q235 and SPHC steels at three casting speeds shows similar characteristics. The values of the RUB at the three casting speeds are all positive in the whole secondary cooling zone, which means that slab broadening existed for Q235 and SPHC steels at these speeds. The RUB changed from one segment to another for the first five segments. The RUB increased and then gradually decreased after reaching its maximum at the fifth and sixth segments. Near the tenth segment, the RUB decreased smoothly; the slab

The simulations of Q235 and SPHC steels produced similar results. The RUB increased with increasing casting speed. For Q235 steel, when the casting speeds were 1.0, 1.1, and 1.2 m/min, the maximum RUB values were 1.44%, 1.88%, and 2.04 %, respectively, and the RUB

from meniscus of a SPHC steel at three casting speeds. (FU JianXun et al. 2011(b))

became completely solidified at this location.

(b)Coefficient of thermal expansion of Q235 and SPHC. (Jian-Xun Fu et al , 2011b)

For continuous caster #2 at Maanshan Iron and Steel Co. Ltd., the slab widths of two steel grades were tracked online at the exit of the caster (the end of the 15th segment); the slab width was measured once per minute. For each grade of steel, measurements were taken for more than 70 minutes. The data are shown in Figures 7(a), and (b), respectively.

Figure 7(a) shows the slab width and the RUB of SPHC steel at various moments. Slab broadening can be clearly seen. The RUB of SPHC steel ranges from 1.4% to 2.4%, with an average of 1.96%. The average RUB is greater than the ratio of linear shrinkage, indicating that the width of the slab after cooling was greater than the top width of the mold. This result shows that slab broadening occurred in the secondary cooling zone.

The slab width changed smoothly except from 45 to 55 min, during which time a sharp trough appears on the RUB curve. In the initial 6 minutes of this period, the RUB decreased to 1.4% from 2.25%, and in the following 4 minutes, the RUB increased to 2.1% from 1.4%. This trough was caused by the changing of the tundish, during which the casting speed decreased sharply, and then quickly recovered to normal; i.e., the change in casting speed caused the change in slab broadening.

Numerical Simulation of Slab Broadening in Continuous Casting of Steel 569

**0.0**

**0.2**

**0.4**

**0.6**

**Casting speed, m/min**

**0.8**

.

*RUB* **of slab**

**Casting speed**

**1.0**

**1.2**

**0 30 60 90 120 150 180 210 240 270 300**

**Time, min**

**5. Effects of width and thickness on slab broadening (FU JianXun et al.** 

One node of the slab was traced and the width was recorded at various positions of the

The calculated RUB of Q235 steel slab with a cross section 2000 mm × 230 mm at speed of 1.0m/min is shown in Figure 9(a). The RUB changes from one segment to another; its value is over 0 throughout the secondary cooling zone, indicating slab broadening. The RUB increases in the first five segments, and then drops down gradually after reaching its maximum in the sixth segment. In the sixth segment, the width of the slab reaches its maximum with a large fluctuation due to the bulging of the slab in the direction of thickness. Figure 9(b) shows the simulated deformation of the slab in this direction. The shell of the slab has low yield strength and high plasticity; thus, the slab at the points contacting the rollers is depressed and bulges at the slit between the two rollers. Similar to

The simulated broadening and bulging of the slab in the sixth segment are shown in Figure 10. There is an obvious correlation between broadening in the width direction and bulging in the thickness direction. The position in the slab where the smallest bulging is observed has the greatest broadening. This is due to the depression of slab in the thickness direction

230-mm-thick slabs of Q235 with various widths were simulated at a casting speed of 1.0 m/min. The RUB values for various segments are shown in Figure 11. It shows that the

secondary cooling zone. The RUB was derived from the calculated width of the slab.

the periodicity of bulging, the width of the slab fluctuates periodically.

**5.2. Effects of slab width on broadening (FU JianXun et al. 2010(b))** 

contributing to slab broadening in the width direction.

**Figure 8.** Relationship between the RUB and casting speed. (FU JianXun et al. 2011(b))

0.0

0.5

1.0

1.5

2.0

*RUB* **of slab, %**

**2010(b))** 

**5.1. Numerical simulation** 

2.5

3.0

3.5

**Figure 7.** (a) Width of slab and RUB for SPHC steel at various moments; (b) Width and RUB for Q235 steel at various moments. (FU JianXun et al. 2011(b))

Figure 7(b) shows the slab width and the RUB for Q235 steel at various times. The RUB for Q235 steel ranges from 0.77% to 2.91%, with an average of 2.04%. There are five sharp corners on the RUB curve for Q235 steel. By comparing the curve with the production process of Q235, it was found that each sharp corner corresponds to an unsteady production stage. The biggest one corresponds to the changing of the tundish, the last one corresponds to the end of casting, and the remaining three correspond to the changing of ladles.

Figure 8 shows the relationship between the RUB and the casting speed for Q235 steel. The shapes of the RUB curve and the casting speed curve are very similar. When the tundish was changed, the casting speed decreased to 0.5 m/min over a 10-minute period and then recovered to normal in 5 minutes; this change formed a sharp trough in the casting speed curve. At nearly the same time, the RUB decreased to 1.91% from 0.77% in 10 minutes and then increased to 2.1% in 5 minutes, producing a sharp trough in the curve. When the ladle was changed, a similar change happened. When the casting speed was maintained at 1.0 m/min, the RUB remained stable at about 2.0%. The RUB is thus closely correlated with casting speed.

There is a small lag between the RUB curve and the casting speed curve in Figure 12. The change in casting speed curve occurred earlier than that in the RUB curve. For example, the casting speed curve exhibits a sharp trough at about 100 minutes; a sharp trough appears in the RUB curve at about 110 minutes. Comparing Figure 7 and Figure 8, it can be seen that the simulation results generally agree with the industrial measurement results.

Numerical Simulation of Slab Broadening in Continuous Casting of Steel 569

**Figure 8.** Relationship between the RUB and casting speed. (FU JianXun et al. 2011(b))
