# Measurement and Model-Based Control of Solidification in Continuous Casting of Steel Billets
## 3 Results
### Dynamic Control of Crater End Position
#### Dynamic Adaptation of Casting Speed and Spray Water Flow Rates  
Two approaches of model-based dynamic control of casting conditions in order to adjust defined target quality parameters for the strands in the billet caster of ESF have been investigated. In the first approach, based on heuristic rules in combination with methods for nested intervals (such as the \"Pegasus\" algorithm, an improved \"Regula Falsi\" method), the casting parameters (such as spray water flow rates and casting speed) are iteratively altered with subsequent model simulation of the resulting stationary three-dimensional temperature field and solidification state; the iteration is stopped, when within defined tolerances, the simulated strand quality parameters agree with the target values. However, this approach with up to 10 iterative simulations of stationary strand states requires too much calculation time (>1 s per iteration) for an online application with cyclic recalculation of the current strand state and adapted control casting parameters each 5-10 s (on the available hardware at ESF). Thus, a second approach has been chosen, based on evaluations of the parameter studies for the ESF billet caster, which have been carried out with appropriate offline model simulations (cf., Section 3.3.1.). The ESF billet caster has only limited space behind the cutting torch. Thus, the adjustment of the crater end position to a defined target value has been identified as a useful dynamic control application. The variation of the crater end position pos_CE with respect to the reference value pos_CE_ref simulated for the defined reference casting conditions (d_pos_CE = pos_CE - pos_CE_ref) under variation of (a) the casting speed (by d_v_c = v_c - v_c_ref) and (b) the spray water flow (by d_SPW = 100 \(\times\) (SPW_i - SPW_i_ref)/SPW_i_ref for each loop i = 1... 4) is shown in Figure 15. The dependency of the crater end position from the casting speed can be expressed by a linear equation:  
\[\text{d\_pos\_CE}=\text{a}\times\text{d\_v\_c} \tag{5}\]  
whereas the dependency of the crater end position from the spray water flow is given by a quadratic equation:  
\[\text{d\_pos\_CE}=\text{c}\times\text{d\_SPW}^{2}-\text{b}\times\text{d\_SPW} \tag{6}\]  
Figure 15: Variation of crater end positions with (**a**) altered casting speed and (**b**) altered spray water flows (with respect to defined reference case) as simulated within offline parameter studies. Equations (5) and (6) are used to calculate the crater end position for casting speeds and spray water flows within a certain range of deviations d_v_c and d_SPW from the defined reference values by the regression formula:  
\[\text{pos\_CE}=\text{pos\_CE\_ref}+\text{c}\times\text{d\_SPW}^{2}\text{-b} \times\text{d\_SPW}+\text{a}\times\text{d\_v\_c} \tag{7}\]  
The reference value for the crater end position at reference casting speed and spray water flows may depend on the initial temperature and composition of the steel powered into the mould. The mould cooling water flow is more or less constant at ESF billet caster and the temperature increase in the mould cooling water scales (more or less linear) with the casting speed, which has been taken into account within the parameter studies leading to Equation (5). With variations of the spray water flow between d_SPW_min = \(-50\%\) and d_SPW_max = \(+50\%\) of the reference flow rates, an adjustment of the crater end position between d_pos_CE_min = \(-1.3\) m and d_pos_CE_max = \(+2.3\) m is achievable (cf., Figure 15b). Beyond this range, the crater end position has to be controlled via the casting speed. Statistical evaluation of a real sequence casted at ESF with a certain range of different casting speeds revealed linear dependencies of the spray water flow rates and the resulting simulated crater end position from the chosen casting speed (cf., Figures 16 and 17).