# Analysis of the Influence of Segmented Rollers on Slab Bulge Deformation
## 2 3D Solidification Model Description
### 3D Solidification Finite Element Model  
The cooling process of the continuous casting slab was a three-dimensional transient heat transfer process with heat conduction, thermal convection, and thermal radiation accompanied by phase transition. The heat transfer differential equation can be expressed as [30]:  
\[\rho c_{p}\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\bigg{(} \lambda\frac{\partial T}{\partial x}\bigg{)}+\frac{\partial}{\partial y}\bigg{(} \lambda\frac{\partial T}{\partial y}\bigg{)}+\frac{\partial}{\partial z}\bigg{(} \lambda\frac{\partial T}{\partial z}\bigg{)}+q \tag{1}\]The axial position \(z\) along the liquid level of crystallizer is related to casting speed \(u\) and time \(t\). The relation function is \(z=ut\). The relation function is substituted into Equation (1) and Equation (2) is shown as follows:  
\[\rho c_{p}u\frac{\partial T}{\partial z}=\frac{\partial}{\partial x}\bigg{(} \lambda\frac{\partial T}{\partial x}\bigg{)}+\frac{\partial}{\partial y}\bigg{(} \lambda\frac{\partial T}{\partial y}\bigg{)}+\frac{\partial}{\partial z}\bigg{(} \lambda\frac{\partial T}{\partial z}\bigg{)}+q \tag{2}\]  
where  
\(\rho\) is the density (\(kg\cdot m^{-3}\)),  
\(c_{p}\) is the specific heat capacity under constant pressure (\(J\cdot(kg\cdot K)^{-1}\)),  
\(T\) is the temperature (\(K\)),  
\(t\) is time (s),  
\(q\) is the internal heat source (\(W\cdot m^{-3}\)),  
\(\lambda\) is the thermal conductivity (\(W\cdot(m\cdot K)^{-1}\)), and  
\(u\) is the casting speed, (\(m\cdot\min^{-1}\)). There is a solid phase zone, a liquid phase zone, and a two-phase zone in the continuous casting process, and Equation (1) can be treated as follows:  
(1) The solid phase and liquid phase can be calculated as:  
\[\rho c_{p}\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\bigg{(} \lambda\frac{\partial T}{\partial x}\bigg{)}+\frac{\partial}{\partial y} \bigg{(}\lambda\frac{\partial T}{\partial y}\bigg{)}+\frac{\partial}{ \partial z}\bigg{(}\lambda\frac{\partial T}{\partial z}\bigg{)} \tag{3}\]  
The heat transfer differential equation of the slab in the solid phase zone and the liquid phase zone is exactly the same Equation (3). The corresponding thermal conductivity can be obtained in the solid phase zone and the liquid phase zone separately. For the thermal conductivity \(\lambda_{L}\) of the liquid phase zone, the forced convection heat transfer was caused due to the flow of the molten steel, which accelerated the elimination of the degree of superheat. In 1967, [PERSON] [31] introduced the effective thermal conductivity for the first time to take the effects of convective heat transfer into account. The method was to treat the liquid phase zone of the slab into a \"quasi-solid\" and convert the convective thermal conductivity of the molten steel into an effective thermal conductivity, which was equivalent to \(n\) times compared with the thermal conductivity of the static molten steel. The value of \(n\) was generally from 2 to 7. Therefore, this method was used to calculate the thermal conductivity of the liquid phase zone in the paper and the value of \(n\) was 5. The relationship between the thermal conductivity of the solid phase zone and the thermal conductivity of the liquid phase zone was as follows:  
\[\lambda_{L}=5\lambda_{S} \tag{4}\]  
where  
\(\lambda_{L}\) is the thermal conductivity of the solid phase zone (\(W\cdot(m\cdot K)^{-1}\)), and  
\(\lambda_{S}\) is the thermal conductivity of the liquid phase zone (\(W\cdot(m\cdot K)^{-1}\)). (2) Since the two-phase zone has the latent heat of solidification, the internal heat source \(q\) in Equation (1) must be taken into consideration.