# Measurement and Model-Based Control of Solidification in Continuous Casting of Steel Billets
## 3 Results  
### Dynamic Temperature and Solidification Model DynSolidCC  
#### 3.1.1 Submodels for Boundary Conditions  
The information about the boundary conditions available from the process data do not allow to distinguish at a certain length below the meniscus the heat transfer across the different strand surfaces, i.e., top, bottom, left, and right sides. Moreover, the DynSolidCC model only considers the mass flow in the casting direction and does not take into account any other fluid dynamics in order to be online-applicable in real time. Thus, symmetric boundary conditions and heat flow densities regarding top and bottom as well as left and right strand surfaces have been assumed for the model simulations and the calculation domain has been restricted to the top left quarter of the strand. The boundary conditions along the strand can be divided into three areas:  
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* Mould area;
* Secondary cooling zones with spray water loops;
* Radiation zone behind spray water loops.  
Figure 3: HMI in ESF control room with results from DynSolidCC functions as well as traffic light system.  
For the mould, one can assume a heat flux q across the surface exponentially decreasing along the casting direction (z) [4; 5; 6]. The maximum heat flux q\({}_{0}\) (at z = 0) can be calculated from the average one. The latter can be estimated from the width W and the length L of the mould, the primary cooling water amount per time w\({}_{\rm P}\) and the temperature increase of this water \(\Delta\)T\({}_{\rm wP}\), which leads to the equation  
\[\rm{q(z)}=q_{0}e^{-\alpha z}\;with\;q_{0}=\frac{a_{c}c_{pw}w_{p}\Delta T_{wp} \alpha}{W(1-e^{-\alpha L})} \tag{1}\]  
where c\({}_{\rm pw}\) is the specific heat of water and a\({}_{\rm c}\) and \(\alpha\) are model parameters. Then, the heat transfer coefficients in the mould amount to  
\[\rm{h(z)}=\frac{q(z)}{T_{surf}-T_{env}} \tag{2}\]  
where T\({}_{\rm surf}\) is the surface temperature of the strand and T\({}_{\rm env}\) is the environment temperature.  
For the secondary cooling zones with spray water amounts per time w\({}_{\rm i}\) in loop i, the heat transfer coefficients can be described by  
\[\rm{h_{i}}=a_{i}\Big{(}b_{i}w_{i}+c_{i}w_{i}^{0,31}\Big{)}+d_{i}w_{i}^{n_{i}}+ f_{i} \tag{3}\]  
where the coefficients a\({}_{\rm i}\), b\({}_{\rm i}\), c\({}_{\rm i}\), d\({}_{\rm i}\), n\({}_{\rm i}\) and f\({}_{\rm i}\) (i = 1, \(\ldots\), number of loops) can be approximately calculated from the caster design data (width, roller distances) and empirical data (comparison with coefficients for known casters) [4; 5; 6].  
For the domain behind the secondary cooling zones, one can assume electromagnetic radiation according to the \"T\"-law\" and an additive term for other heat transport across the strand surface, leading to a heat transfer coefficient  
\[\rm{h}=a_{r}\sigma\;\frac{T_{surf}^{4}-T_{env}}{T_{surf}-T_{env}}+b_{r} \tag{4}\]  
where \(\sigma\) = 5.67 \(\times\) 10\({}^{-8}\) W/(m\({}^{2}\)K) is the Stefan-Boltzmann constant and a\({}_{r}\) and b\({}_{r}\) are model parameters.  
The parameters used in the boundary condition models given in Equations (1)-(4) have been adapted to the specific casting process and machine at ESF based on the above-mentioned measurement campaigns.