High Quality Steel Casting by Using Advanced Mathematical Methods

metals Article High Quality Steel Casting by Using Advanced Mathematical Methods Tomas Mauder * and Josef Stetina Energy Institute, Brno University of Technology, 616 69 Brno, Technicka 2, Czech Republic; [email protected].cz *Correspondence: [email protected].cz; Tel.: +420-541-14-3252 Received: 15 November 2018; Accepted: 30 November 2018; Published: 4 December 2018   Abstract: The main concept of this paper is to utilize advanced numerical modelling techniques with self-regulation algorithm in order to reach optimal casting conditions for real-time casting control. Fully 3-D macro-solidification model for the continuous casting (CC) process and an original fuzzy logic regulator are combined. The fuzzy logic (FL) regulator reacts on signals from two data inputs, the temperature field and the historical steel quality database. FL adjust the cooling intensity as a function of casting speed and pouring temperature. This approach was originally designed for the special high-quality high-additive steel grades such as higher strength grades, steel for acidic environments, steel for the offshore technology and so forth. However, mentioned approach can be also used for any arbitrary low-carbon steel grades. The usability and results of this approach are demonstrated for steel grade S355, were the real historical data from quality database contains approximately 2000 heats. The presented original solution together with the large steel quality databases can be used as an independent CC prediction control system. Keywords: continuous casting; fuzzy logic; optimal cooling; steel quality prediction 1. Introduction The continuous casting technology is a well-known predominant process how the steel is produced in the world. CC is already fully-grown technology and successfully casting million tonnes of classic low-carbon and low-alloy steel grades per year. In spite of this fact, casting of special high-strength grades, steels for acidic environment, steels for the offshore technology, high alloyed tool steels, can be still challenging in order to ensure a constant steel quality through the whole casting process. Flick and Stoiber [ 1 ] described present issues and future trends in the CC technology. The quality of the steel is still a discussed topic. In CC the solid shell is permanently subjected to thermal and mechanical stresses and it can give rise to crack formation, see Birat et al. [ 2 ]. Figure 1 shows CC installation and mechanical tensile/compression stresses in specific locations. The rejected slabs, by the quality control system, which need to be scraped is very uneconomic. The breakdown situation caused by a low quality of these steels could have a catastrophic effect on material and human losses. This quality issue can be handled by using the advanced numerical modelling methods, optimization-regulation techniques and statistical evaluations of the real casting data. Historically, there are many papers which combine the numerical modelling and optimizationregulation approaches, such as Santos et al. [ 3 ] who applied the genetic algorithm, Zhemping et al. [4] verified the usage of the ant colony algorithm, Zheng et al. [ 5 ] attempted to use the swarm optimization on 2-D solidification model, Mauder and Novotny [ 6 ] showed the possibility of using classical mathematical programming method and compared the results with simple Nelder-Mead algorithm, Ivanova [ 7 ] constructed basic predictive control algorithm for 2-D solidification model and so forth. Rao et al. [ 8 ] publish a comprehensive review dealing with parameters optimization of selected casting processes. Unfortunately, these works are far from the use on the real casting process, Metals 2018,8, 1019; doi:10.3390/met8121019 www.mdpi.com/journal/metals Metals 2018,8, 1019 2 of 13 because they often calculate very simplified solidification models (simple geometry, 2-D mesh, simple boundary conditions, contain a small steel database, etc.) and it is proven that the numerical results of these models in compare with the 3-D fine-mesh validated solidification models are poor, see Mosayebidorcheh and Bandpy [ 9 ]. Moreover, the optimization algorithms are often based on black-box approach that generally needs a large number of optimization iterations before the optimal solution is found. This is not a problem in the case of simple solidification models, which calculates the temperature field very fast. However, in the case of the complex 3D numerical simulations it is not possible to use these optimization algorithms for the real time control. Metals 2018, 8, x FOR PEER REVIEW 2 of 13 Figure 1. Scheme of the continuous casting and mechanical stresses during bending and straightening. 1—tundish; 2—mould; 3—nozzle; 4—cooling circuit; 5—roller. Historically, there are many papers which combine the numerical modelling and optimizationregulation approaches, such as Santos et al. [3] who applied the genetic algorithm, Zhemping et al. [4] verified the usage of the ant colony algorithm, Zheng et al. [5] attempted to use the swarm optimization on 2-D solidification model, Mauder and Novotny [6] showed the possibility of using classical mathematical programming method and compared the results with simple Nelder-Mead algorithm, Ivanova [7] constructed basic predictive control algorithm for 2-D solidification model and so forth. Rao et al. [8] publish a comprehensive review dealing with parameters optimization of selected casting processes. Unfortunately, these works are far from the use on the real casting process, because they often calculate very simplified solidification models (simple geometry, 2-D mesh, simple boundary conditions, contain a small steel database, etc.) and it is proven that the numerical results of these models in compare with the 3-D fine-mesh validated solidification models are poor, see Mosayebidorcheh and Bandpy [9]. Moreover, the optimization algorithms are often based on black-box approach that generally needs a large number of optimization iterations before the optimal solution is found. This is not a problem in the case of simple solidification models, which calculates the temperature field very fast. However, in the case of the complex 3D numerical simulations it is not possible to use these optimization algorithms for the real time control. This paper describes both, the original 3-D solidification model, the so-called Brno Dynamic Solidification Model ® (BrDSM) and advanced optimal control algorithm based on the fuzzy logic (FLBrDSM). This unique combination represents a tool for achievement of high steel quality products. Besides of the numerical and fuzzy regulation models, special attention must be concentrated to proper setting of thermophysical parameters of investigated steel grade, experimental measurement of boundary conditions and statistical evaluations of the real casting data. Presented approach is demonstrated on the real radial slab caster with twelve cooling loops in the secondary cooling for the special grade of steel S355. This steel was selected expediently because it is mainly used for shipbuilding projects, marine mechanical systems and deep-water ocean offshore structural projects were the quality of steel S355 plates is essential. The mechanical properties for these grades of steel are specified by the European Standards. 2. Solidification Model—BrDSM The core of presented approach is the 3-D transient solidification model. The computing precision, accuracy, robustness and fast computational times are essential. In order to reach these specifications a suitable numerical scheme should be selected. Numerical mesh independence tests as well as tests of the calculation speed on different CPU/GPU and mainly a properly validation on the real casting data were made. Figure 1. Scheme of the continuous casting and mechanical stresses during bending and straightening. 1—tundish; 2—mould; 3—nozzle; 4—cooling circuit; 5—roller. This paper describes both, the original 3-D solidification model, the so-called Brno Dynamic Solidification Model ® (BrDSM) and advanced optimal control algorithm based on the fuzzy logic (FL-BrDSM). This unique combination represents a tool for achievement of high steel quality products. Besides of the numerical and fuzzy regulation models, special attention must be concentrated to proper setting of thermophysical parameters of investigated steel grade, experimental measurement of boundary conditions and statistical evaluations of the real casting data. Presented approach is demonstrated on the real radial slab caster with twelve cooling loops in the secondary cooling for the special grade of steel S355. This steel was selected expediently because it is mainly used for shipbuilding projects, marine mechanical systems and deep-water ocean offshore structural projects were the quality of steel S355 plates is essential. The mechanical properties for these grades of steel are specified by the European Standards. 2. Solidification Model—BrDSM The core of presented approach is the 3-D transient solidification model. The computing precision, accuracy, robustness and fast computational times are essential. In order to reach these specifications a suitable numerical scheme should be selected. Numerical mesh independence tests as well as tests of the calculation speed on different CPU/GPU and mainly a properly validation on the real casting data were made. The transient 3-D solidification model is based on Fourier-Kirchhoff Equation [ 2 ] which can be expressed as: ∂(hρ) ∂τ +vcast ∂(hρ) ∂z=∇ · ke f f ∇T, (1) Metals 2018,8, 1019 3 of 13 with boundary conditions: T(x,y,z)z=0=Tpouring the meniscus, (2) −λ∂T ∂n=0 the exit area, (3) −λ∂T ∂n= . mwatercwater(Tout −Tin) Smold in the mould, (4) −λ∂T ∂n=π(l/2)drol Srol htcrol(Trol −Tamb)+σεT4 rol −T4 ambbeneath the rollers, (5) −λ∂T ∂n=htc(T−Twater)+σεT4−T4 waterbeneath the spray cooling, (6) −λ∂T ∂n=0.84(T−Tamb)4/3+σεT4−T4 ambwithin the solidified shell free surface. (7) k eff is the effective thermal conductivity (W/mK); Tis the temperature (K); T pouring is the pouring temperature (K); his the specific enthalpy (J/kg); ρ is the density (kg/m 3 ); τ is the time (s); v cast is the casting speed (m/min); zis the direction of casting (m), T rol is the roller temperature; T water is the cooling water temperature; T amb is the ambient temperature; . mwater is the mass water flow in the mould (kg/s); c water is the specific heat capacity of water (J/kgK); htc is the heat transfer coefficient beneath spraying surface (W/m 2 K); σ is the Stefan-Boltzman constant (W/m 2 K 4 ) and ε is the emissivity of the slab surface (-). The latent heat released during the solidification in Equation (1) is substituted by the enthalpy h and can be expressed as: h= T Z 0c(ζ)−∆H∂fs ∂ζ dζ, (8) where ∆ His the latent heat (J/kg) and f s is the solid fraction (-). The Enthalpy method was used for modelling the solidification process, see Mauder et al. [ 10 ]. The Enthalpy method is robust method because it ensures energy conservation and there is no discontinuity at either the liquidus or the solidus temperatures because the solidification/melting path is characterized strictly by decreasing/increasing enthalpy. In the Equation (1) the enthalpy is calculated in the first step as the primary variable and the temperature is calculated from a defined enthalpy-temperature relationship in the second step Equation (8). In the presented solidification model, Equation (8) is substituted by an enthalpy-temperature function obtained by the InterDendritic Solidification (IDS) package created by the Miettinen [ 11 ]. The IDS allows calculation of enthalpy, thermal conductivity, density and other thermophysical parameters as a function of temperature from 1600 ◦C–0 ◦C. Fluid flow of the liquid steel can be calculated by the computational fluid dynamics (CFD) techniques. However, computational times are even with the use of new modern modelling approaches far from the real duration of the simulated process. The influence of heat transfer by the fluid flow is account in the effective thermal conductivity term k eff . The thermal conductivity is increased by the flow of liquid steel at different distances from the meniscus, where k eff is represented by Zhang et al. [12] as follows: ke f f =                      k T ≤Tsol 4k+3k(T−Tliq) (Tpouring−Tliq)T≥Tliq ∧0≤z≤1 m k+3k(T−Tsol) (Tliq−Tsol )Tsol ≥T≥Tliq ∧0≤z≤1 m k+k(T−Tliq) (Tpouring−Tliq)T≥Tliq ∧1m ≤z≤3 m k z ≥3 m, (9) Metals 2018,8, 1019 4 of 13 where kis the thermal conductivity (W/mK); T sol is the temperature of solidus (K); T liq is the temperature of liquidus (K); and zis the distance from the meniscus (m). The heat transfer in the mould region described by Equation (4) is calculated from the heat balance between heat removal from the mould walls and internal water cooling in-out temperature difference. The real data comes directly from the measurement of cooling water temperatures and flow rates, which are affected by the casting speed and steel carbon content. The situation beneath the rollers Equation (5) also comes from the heat balance assumption. The removed heat from the slab surface is equal to the heat, which is radiated to the surroundings by the rollers. In the case of internal cooled rollers, Javurek et al. [ 13 ] published a paper on how to determine boundary conditions in the case of dry casting. In the spraying area, the strand is cooled by the water or water–air mixture sprays. Nozzle parameters like air and water flow, nozzle position and impact angles have an effect on the cooling efficiency, the htc from the Equation (6) respectively. Totten et al. [ 14 ] showed several empirical formulas describing how to deal with the heat transfer coefficient beneath the nozzle. However, these empirical formulas include many constants and parameters and their correct determination for a particular cooling setup is difficult. Proper determination of boundary conditions is crucial in case to obtain a real results, see Lopez et al. [15]. The advantage of the model discussed in this paper is that it obtains its heat transfer coefficients from measurements of the spraying characteristics of all nozzles used by the caster on a so-called hot plate in the experimental laboratory, according to Raudensky et al. [ 16 ]. Thus, the model takes htc as a function of water flow, casting speed, surface temperature, air pressure and can be expressed mathematically as: htc =f. mwater,vcast,Tsur f ace,pair. (10) 3. Numerical Formulation and Massive Parallelization There are several discretization schemes how to solve the Equations (1)–(7). After large-scale numerical studies two numerical schemes proved to be good candidates for solving the case of 3-D transient heat transfer problem with solidification phenomena, Simple Explicit (SE) and Alternating Direction Implicit (ADI), see Mauder et al. [ 10 ]. The ADI scheme proposed by Douglas and Gunn is unconditionally stable and retains the second-order accuracy when applied to 3-D problems. However, ADI scheme has very limited possibility of parallelization on multiple CPU and massive parallelization on GPU is out of question. In the case of complex 3-D solidification-optimization models, the requirement for the real time control can be reached only by the use of the massive parallelization techniques. In Table 1there are computation times for ADI and SE schemes tested on three different types of meshes (coarse-mesh with 10 000 nodes, fine-mesh with 125,000 nodes and the very-fine mesh with 1,000,000 nodes) on Intel(R) Core(TM) i7-3770 CPU @ 3.4GHz 16GB RAM. More information about efficiency, robustness and accuracy of the examined numerical schemes can be found in Mauder et al. [10]. Table 1. Computation time for non-parallel and parallel solution (s). Numerical Scheme Computational Time (s) Coarse mesh Fine-mesh Very-fine mesh SE—1 CPU 11.54 887.14 54,362.12 SE—12 CPU 135.28 1122.13 35,124.54 ADI—1 CPU 41.81 714.73 36,210.41 ADI—12 CPU 58.61 824.32 32,416.32 SE—GPU 18.67 97.72 972.49 This is the reason why the numerical core in presented solidification model is created by SE scheme. On the one hand, SE has its limitation in the form of stability condition, thus for fine meshes Metals 2018,8, 1019 5 of 13 the time step has to be very small. However, due to the use of massive parallelization on GPU this issue is compensated. The calculation on GPU is more than 50 times faster for very-fine meshes. The SE scheme applied to Equation (1) can be in the form: hn+1 i,j,kρn+1 i,j,k=hn i,j,kρn i,j,k+∆τh. Qx +. Qy +. Qzi−∆τvcast hn i,j,k+1ρn i,j,k+1−hn i,j,kρn i,j,k ∆zk−1 , (11) where: . QΨ=20.5 kn ξ+1 +0.5 kn ξ−1Tn ξ+1−Tn ξ ∆Ψξ−0.5 kn ξ−1 +0.5 kn ξ−1Tn ξ−Tn ξ−1 ∆Ψξ−1 ∆Ψξ+∆Ψξ−1Ψ∈{x,y,z}ξ∈{i,j,k}.(12) The index n+ 1 is associated to the future time; nis the index corresponding to the actual time; ∆τ is the increment of the time; x,yand zare the directions; ∆Ψ are increments in the directions and ξ are the positions. The time step ∆τhas to fulfil a well-known stability condition: ∆τ≤1 2k(T) ρ(T)c(T)∑Ψ1 ∆Ψ2+vcast ∆z .Ψ∈{x,y,z}(13) The numerical model has a non-equidistant mesh in all directions. The reason for this is that the largest temperature gradients are near the surface. In axis z(direction of casting) the nodes are adapted to the real rollers and nozzles positions to get more accurate determination of boundary conditions in secondary cooling zone. Mesh independence test in Figure 2indicates when the convergence is achieved. In xaxis is the number of elements, in y-left axis is the temperature in several points on the top surface (distance from meniscus 1, 2, 4, 10, 15, 20 m) and in the y-right axis is the value of the metallurgical length. This study shows that the number of elements should be at least 2 500,000. Finer mesh do not affected the results significantly. These results with respect the results obtained in Table 1 show again the needs of SE GPU parallel scheme in order to calculate complex 3-D solidification model in the real time. Metals 2018, 8, x FOR PEER REVIEW 5 of 13 Numerical Scheme Coarse mesh Fine-mesh Very-fine mesh SE—1 CPU 11.54 887.14 54,362.12 SE—12 CPU 135.28 1122.13 35,124.54 ADI—1 CPU 41.81 714.73 36,210.41 ADI—12 CPU 58.61 824.32 32,416.32 SE—GPU 18.67 97.72 972.49 This is the reason why the numerical core in presented solidification model is created by SE scheme. On the one hand, SE has its limitation in the form of stability condition, thus for fine meshes the time step has to be very small. However, due to the use of massive parallelization on GPU this issue is compensated. The calculation on GPU is more than 50 times faster for very-fine meshes. The SE scheme applied to Equation (1) can be in the form: ℎ,, ,, =ℎ,, ,, +∆󰇗+󰇗+󰇗−∆ ,, ,, ,, ,,  ∆ , (11) where: 󰇗=2󰇧0.5  +0.5 󰇨 − ∆−󰇧0.5  +0.5 󰇨−  ∆ ∆  +∆   ∈󰇝,,󰇞 ∈󰇝,  ,󰇞. (12) The index n + 1 is associated to the future time; n is the index corresponding to the actual time; Δτ is the increment of the time; x, y and z are the directions; ΔΨ are increments in the directions and ξ are the positions. The time step Δτ has to fulfil a well-known stability condition: ∆≤ 1 2󰇛󰇜 󰇛󰇜󰇛󰇜󰇡∑1 ∆ 󰇢+ ∆ . ∈󰇝,,󰇞 (13) The numerical model has a non-equidistant mesh in all directions. The reason for this is that the largest temperature gradients are near the surface. In axis z (direction of casting) the nodes are adapted to the real rollers and nozzles positions to get more accurate determination of boundary conditions in secondary cooling zone. Mesh independence test in Figure 2 indicates when the convergence is achieved. In x axis is the number of elements, in y-left axis is the temperature in several points on the top surface (distance from meniscus 1, 2, 4, 10, 15, 20 m) and in the y-right axis is the value of the metallurgical length. This study shows that the number of elements should be at least 2 500,000. Finer mesh do not affected the results significantly. These results with respect the results obtained in Table 1 show again the needs of SE GPU parallel scheme in order to calculate complex 3D solidification model in the real time. Figure 2. Temperatures and metallurgical length for different mesh densities. Figure 2. Temperatures and metallurgical length for different mesh densities. The mesh decomposition is sketched out in Figure 3. The primary mesh is created on CPU, then is divided into many sub-meshes (regions). The boundary conditions and initial condition are set on CPU. Further, numerical regions are separately send to the GPU units where Equations (10) and (11) are calculated. After the temperatures in future time step are calculated, the values in separated regions are send back to CPU where boundary conditions are recalculated and again send to the GPU. This loop is repeated until final temperature field is reached. Metals 2018,8, 1019 6 of 13 Metals 2018, 8, x FOR PEER REVIEW 6 of 13 The mesh decomposition is sketched out in Figure 3. The primary mesh is created on CPU, then is divided into many sub-meshes (regions). The boundary conditions and initial condition are set on CPU. Further, numerical regions are separately send to the GPU units where Equations (10) and (11) are calculated. After the temperatures in future time step are calculated, the values in separated regions are send back to CPU where boundary conditions are recalculated and again send to the GPU. This loop is repeated until final temperature field is reached. Figure 3. Numerical mesh CPU-GPU decomposition. 4. Fuzzy Logic Regulator—FL-BrDSM The BrDSM can calculate the temperature distribution of the strand in real time for given casting parameters such as the initial temperature distribution, intensity of cooling, speed of casting and so forth. The problem is how to set input parameters in the optimal way to get a required (optimal) temperature distribution. This is referred to as an inverse problem. For this reason, supervisionsystem/optimal-regulator based on fuzzy logic has been created. The FL regulator treats the solidification model as a black-box and when the FL regulator changes input parameters such as casting speed and intensity of cooling, the black-box returns the final temperature field for new input parameters. Based on the temperature field the FL regulator adjusts input parameters and repeats the process in a closed-loop until the optimal solution is found. Several regulation approaches have been tested in the past, such as PID regulation, FL regulation, Model Predictive Control (MPC) regulation and their combinations. The most promising regulation outcome for dynamic changes in process parameters was reached by using a combination of MPC and FL regulation, see Stetina et al. [17]. This approach generally requires a large number of iterations because several future scenarios are calculated. For the real time simulation, this is possible only with use of GPU solidification model. The most important parameters for the MPC/FL regulator are so called optimal surface temperature intervals. These temperature intervals should guarantee smooth temperature (cooling) profiles and high quality of final steel with zero or minimum surface defects. The optimal cooling strategy is to keep the surface temperatures in these intervals, see Figure 4. These intervals are set by the user and they are distinct for different grades of steel. In Section 5 is discussed how to set these intervals. Figure 3. Numerical mesh CPU-GPU decomposition. 4. Fuzzy Logic Regulator—FL-BrDSM The BrDSM can calculate the temperature distribution of the strand in real time for given casting parameters such as the initial temperature distribution, intensity of cooling, speed of casting and so forth. The problem is how to set input parameters in the optimal way to get a required (optimal) temperature distribution. This is referred to as an inverse problem. For this reason, supervision-system/optimal-regulator based on fuzzy logic has been created. The FL regulator treats the solidification model as a black-box and when the FL regulator changes input parameters such as casting speed and intensity of cooling, the black-box returns the final temperature field for new input parameters. Based on the temperature field the FL regulator adjusts input parameters and repeats the process in a closed-loop until the optimal solution is found. Several regulation approaches have been tested in the past, such as PID regulation, FL regulation, Model Predictive Control (MPC) regulation and their combinations. The most promising regulation outcome for dynamic changes in process parameters was reached by using a combination of MPC and FL regulation, see Stetina et al. [ 17 ]. This approach generally requires a large number of iterations because several future scenarios are calculated. For the real time simulation, this is possible only with use of GPU solidification model. The most important parameters for the MPC/FL regulator are so called optimal surface temperature intervals. These temperature intervals should guarantee smooth temperature (cooling) profiles and high quality of final steel with zero or minimum surface defects. The optimal cooling strategy is to keep the surface temperatures in these intervals, see Figure 4. These intervals are set by the user and they are distinct for different grades of steel. In Section 5is discussed how to set these intervals. Metals 2018, 8, x FOR PEER REVIEW 7 of 13 Figure 4. Optimal temperature intervals. FL regulator extracts temperatures in the regulation points from the computed result (output from BrDSM) and by compare them with the prescribed temperature values to determine their errors. With all the temperature information, the FL regulator infers modifications for each cooling circuits. The temperatures in the regulation points are influenced by all previous cooling circuits, which should be taken account. The closer cooling circuit has a bigger influence on the regulation point, thus artificial parameter so call impact parameter in the range from 0 to 10 is created. The regulator evaluates the errors and impacts by the fuzzy rules. These rules give the value of modification for each cooling circuit. One circuit can get several different modifications from following control points proportional to the impact. Some modifications can require increase of cooling, the other decrease of cooling. The algorithm uses the final value for modification of one cooling loop as a sum of these values. Membership functions for all fuzzy statements are trapezoidal-shaped uniformly distributed through the corresponding interval. These intervals are professionally/expertly set based on caster geometry and another caster parameters. As a defuzzification method, the standard centre of gravity was chosen. The combination of the FL regulator with the GPU solidification model can predict future temperature states and it works like the MPC system. The detailed description of the FL regulator including linguistic variables and linguistic rules, input and output data and so forth, is described in Mauder et al. [18]. 5. Steel S355 and Real Casting Data A frequent occurrence of surface defects in slabs of the cast steel grade S355 was the reason for statistical and numerical investigation. The importance of high quality of the steel S355 was mentioned in the introduction. The chemical composition is listed in Table 2. Table 2. Chemical composition of steel S355. Weight Fraction Ni Mn Mo Si Nb Ti Cu wt% max 0.300 1.400– 1.550 max 0.080 0.5 max 0.060 max 0.020 max 0.200 Weight Fraction V Al P C Cr S Ca wt% max 0.020 0.020– 0.060 0.030 0.160– 0.180 max 0.200 0.020 0.002 Figure 4. Optimal temperature intervals. Metals 2018,8, 1019 7 of 13 FL regulator extracts temperatures in the regulation points from the computed result (output from BrDSM) and by compare them with the prescribed temperature values to determine their errors. With all the temperature information, the FL regulator infers modifications for each cooling circuits. The temperatures in the regulation points are influenced by all previous cooling circuits, which should be taken account. The closer cooling circuit has a bigger influence on the regulation point, thus artificial parameter so call impact parameter in the range from 0 to 10 is created. The regulator evaluates the errors and impacts by the fuzzy rules. These rules give the value of modification for each cooling circuit. One circuit can get several different modifications from following control points proportional to the impact. Some modifications can require increase of cooling, the other decrease of cooling. The algorithm uses the final value for modification of one cooling loop as a sum of these values. Membership functions for all fuzzy statements are trapezoidal-shaped uniformly distributed through the corresponding interval. These intervals are professionally/expertly set based on caster geometry and another caster parameters. As a defuzzification method, the standard centre of gravity was chosen. The combination of the FL regulator with the GPU solidification model can predict future temperature states and it works like the MPC system. The detailed description of the FL regulator including linguistic variables and linguistic rules, input and output data and so forth, is described in Mauder et al. [18]. 5. Steel S355 and Real Casting Data A frequent occurrence of surface defects in slabs of the cast steel grade S355 was the reason for statistical and numerical investigation. The importance of high quality of the steel S355 was mentioned in the introduction. The chemical composition is listed in Table 2. Table 2. Chemical composition of steel S355. Weight Fraction Ni Mn Mo Si Nb Ti Cu wt% max 0.300 1.400–1.550 max 0.080 0.5 max 0.060 max 0.020 max 0.200 Weight Fraction V Al P C Cr S Ca wt% max 0.020 0.020–0.060 0.030 0.160–0.180 max 0.200 0.020 0.002 Three typical defects were found: transverse facial cracks, star cracks and longitudinal facial cracks. The most common defects were the star cracks. These defects primarily appear at the top surface of the strand and possible causes for these defects are hard cooling and tensile stresses at the straightening area [ 2 ]. Therefore, the top surface of the strand and the temperatures at the straightening area were investigated. The real casting data were statistically evaluated from approximately 2000 heats cast in 2011 and 2012 in EVRAZ VITKOVICE STEEL machinery in the Czech Republic. The evaluation of the statistical hypothesis shows that the surface temperature in the unbending point significantly influences the surface quality of slabs. The heats were divided into two groups. Heats where surface defects were found and heats without surface defects. The surface temperature was measured by a pyrometer at the straightening area. The results in a form of histogram are shown in Figure 5. The heats without surface defects were fitted by the Gauss curve to obtain optimal temperature intervals at unbending point. Metals 2018,8, 1019 8 of 13 Metals 2018, 8, x FOR PEER REVIEW 8 of 13 Three typical defects were found: transverse facial cracks, star cracks and longitudinal facial cracks. The most common defects were the star cracks. These defects primarily appear at the top surface of the strand and possible causes for these defects are hard cooling and tensile stresses at the straightening area [2]. Therefore, the top surface of the strand and the temperatures at the straightening area were investigated. The real casting data were statistically evaluated from approximately 2000 heats cast in 2011 and 2012 in EVRAZ VITKOVICE STEEL machinery in the Czech Republic. The evaluation of the statistical hypothesis shows that the surface temperature in the unbending point significantly influences the surface quality of slabs. The heats were divided into two groups. Heats where surface defects were found and heats without surface defects. The surface temperature was measured by a pyrometer at the straightening area. The results in a form of histogram are shown in Figure 5. The heats without surface defects were fitted by the Gauss curve to obtain optimal temperature intervals at unbending point. Figure 5. Surface temperatures in unbending point with and without defects. Hypothesis tests show that we cannot reject the hypothesis on the 5% level of significance stating that the surface temperature at the straightening area influences the presence of surface defects. This leads back to the following idea: if we will keep the surface temperature in some range, the occurrence of surface defects would be minimized. The optimal temperature intervals at unbending point were set according to the statistical results from heats without defects 916.03 ± 6.89 °C. There are three pyrometers distributed on the examined caster. First one at end of the mould, the second one at unbending point (Figure 5) and the last one at the end of tertiary cooling zone. The most significant area is at unbending point from the perspective of cracks. However, smooth decreasing of surface temperatures through the casting process has positive influence on the final steel quality. Thus, the optimal temperature intervals for the fuzzy regulator were obtained by the curve fitting of these three points using following function: =+, (14) where y is the temperature, x is distance from the meniscus and constant a, b and c were found by using curve fitting tool. These constants have to be found for each grade of steel separately. Optimal temperature intervals obtained this way are used as the input parameters for FL-BrDSM. Thermophysical properties for the steel grade S355 as functions of the temperature were calculated using the InterDendritic Solidification (IDS) thermodynamic-kinetic package (version 1.3.1, Helsinki University of Technology, Helsinki, Finland). The results are presented in Figure 6a. The solidification model of the IDS is a so-called “grey box”, that is, it combines empirical or semiemprical submodels with physically conceived submodels. The IDS model has been created at Aalto University in Helsinki [11] and is further developed, see Louhenkilpi et al. [19]. The IDS model consists of two main submodels for the simulation of interdendritic solidification (solves solidification from liquidus temperature to 1000 °C, i.e., the formation of, for example, ferrite or Figure 5. Surface temperatures in unbending point with and without defects. Hypothesis tests show that we cannot reject the hypothesis on the 5% level of significance stating that the surface temperature at the straightening area influences the presence of surface defects. This leads back to the following idea: if we will keep the surface temperature in some range, the occurrence of surface defects would be minimized. The optimal temperature intervals at unbending point were set according to the statistical results from heats without defects 916.03 ±6.89 ◦C. There are three pyrometers distributed on the examined caster. First one at end of the mould, the second one at unbending point (Figure 5) and the last one at the end of tertiary cooling zone. The most significant area is at unbending point from the perspective of cracks. However, smooth decreasing of surface temperatures through the casting process has positive influence on the final steel quality. Thus, the optimal temperature intervals for the fuzzy regulator were obtained by the curve fitting of these three points using following function: y=xab+c, (14) where yis the temperature, xis distance from the meniscus and constant a,band cwere found by using curve fitting tool. These constants have to be found for each grade of steel separately. Optimal temperature intervals obtained this way are used as the input parameters for FL-BrDSM. Thermophysical properties for the steel grade S355 as functions of the temperature were calculated using the InterDendritic Solidification (IDS) thermodynamic-kinetic package (version 1.3.1, Helsinki University of Technology, Helsinki, Finland). The results are presented in Figure 6a. The solidification model of the IDS is a so-called “grey box”, that is, it combines empirical or semi-emprical submodels with physically conceived submodels. The IDS model has been created at Aalto University in Helsinki [ 11 ] and is further developed, see Louhenkilpi et al. [ 19 ]. The IDS model consists of two main submodels for the simulation of interdendritic solidification (solves solidification from liquidus temperature to 1000 ◦ C, i.e., the formation of, for example, ferrite or austenite) and simulation of solid state austenability (solves solidification from 1000 ◦ C to temperature, 25 ◦ C, that is, the formation of proeutectoid ferrite, cementite, perlite, bainite and martensite. The model also supports basic calculations of properties that influence the strength behaviour of steel and can serve as a basis for predictive crack criteria, see Figure 6b). Metals 2018,8, 1019 9 of 13 Metals 2018, 8, x FOR PEER REVIEW 9 of 13 austenite) and simulation of solid state austenability (solves solidification from 1000 °C to temperature, 25 °C, that is, the formation of proeutectoid ferrite, cementite, perlite, bainite and martensite. The model also supports basic calculations of properties that influence the strength behaviour of steel and can serve as a basis for predictive crack criteria, see Figure 6b). (a) (b) Figure 6. Temperature dependent physical properties for the steel grade S355, (a) thermophysical properties; (b) Mechanical properties. 6. Results and Discussion The input parameters for the FL-BrDSM were: the thermophysical properties of steel grade S355 (from IDS); optimal surface temperature intervals (from statistical evaluation); speed of casting (set by the user); meniscus temperature (from historical/actual casting data in case of offline/online simulation); heat fluxes at the mould (from historical/actual casting data as a function of casting speed); and maximal and minimal water flows at the secondary cooling (from the caster specification). The output parameters for the FL-BrDSM were the temperature field (from BrDSM) and the optimal cooling intensity at the secondary cooling zone (from the FL regulator). The geometry of the investigated CC machine is as follows: a mould length of 900 mm; a radius of 8000 mm; a length of secondary cooling after an unbending point of 8500 mm; length of tertiary cooling of 2000 mm. The secondary cooling zone is divided into 12 independent cooling loops according to Figure 7. For more detailed caster description see Mauder et al. [18]. Figure 7. Position of cooling loops. Firstly, the FL regulator was tested for static process parameters. The optimal temperature field for the average casting speed 1.7 m/min is shown in Figure 8. The cross-section size of the slab was 1530 mm × 250 mm and the numerical mesh was created from over 1.5 million nodes. Because the CC is a dynamic process and the casting speed varies in time, it is necessary to keep surface temperatures at constant values in order to achieve a high steel quality. Thus, optimization was carried out for two different casting speeds in order to obtain the optimal relationship between cooling intensities and the casting speed, see Table 3. These results can be used for the real CC process in case of steel grade S355 casting. Figure 6. Temperature dependent physical properties for the steel grade S355, ( a ) thermophysical properties; (b) Mechanical properties. 6. Results and Discussion The input parameters for the FL-BrDSM were: the thermophysical properties of steel grade S355 (from IDS); optimal surface temperature intervals (from statistical evaluation); speed of casting (set by the user); meniscus temperature (from historical/actual casting data in case of offline/online simulation); heat fluxes at the mould (from historical/actual casting data as a function of casting speed); and maximal and minimal water flows at the secondary cooling (from the caster specification). The output parameters for the FL-BrDSM were the temperature field (from BrDSM) and the optimal cooling intensity at the secondary cooling zone (from the FL regulator). The geometry of the investigated CC machine is as follows: a mould length of 900 mm; a radius of 8000 mm; a length of secondary cooling after an unbending point of 8500 mm; length of tertiary cooling of 2000 mm. The secondary cooling zone is divided into 12 independent cooling loops according to Figure 7. For more detailed caster description see Mauder et al. [18]. Metals 2018, 8, x FOR PEER REVIEW 9 of 13 austenite) and simulation of solid state austenability (solves solidification from 1000 °C to temperature, 25 °C, that is, the formation of proeutectoid ferrite, cementite, perlite, bainite and martensite. The model also supports basic calculations of properties that influence the strength behaviour of steel and can serve as a basis for predictive crack criteria, see Figure 6b). (a) (b) Figure 6. Temperature dependent physical properties for the steel grade S355, (a) thermophysical properties; (b) Mechanical properties. 6. Results and Discussion The input parameters for the FL-BrDSM were: the thermophysical properties of steel grade S355 (from IDS); optimal surface temperature intervals (from statistical evaluation); speed of casting (set by the user); meniscus temperature (from historical/actual casting data in case of offline/online simulation); heat fluxes at the mould (from historical/actual casting data as a function of casting speed); and maximal and minimal water flows at the secondary cooling (from the caster specification). The output parameters for the FL-BrDSM were the temperature field (from BrDSM) and the optimal cooling intensity at the secondary cooling zone (from the FL regulator). The geometry of the investigated CC machine is as follows: a mould length of 900 mm; a radius of 8000 mm; a length of secondary cooling after an unbending point of 8500 mm; length of tertiary cooling of 2000 mm. The secondary cooling zone is divided into 12 independent cooling loops according to Figure 7. For more detailed caster description see Mauder et al. [18]. Figure 7. Position of cooling loops. Firstly, the FL regulator was tested for static process parameters. The optimal temperature field for the average casting speed 1.7 m/min is shown in Figure 8. The cross-section size of the slab was 1530 mm × 250 mm and the numerical mesh was created from over 1.5 million nodes. Because the CC is a dynamic process and the casting speed varies in time, it is necessary to keep surface temperatures at constant values in order to achieve a high steel quality. Thus, optimization was carried out for two different casting speeds in order to obtain the optimal relationship between cooling intensities and the casting speed, see Table 3. These results can be used for the real CC process in case of steel grade S355 casting. Figure 7. Position of cooling loops. Firstly, the FL regulator was tested for static process parameters. The optimal temperature field for the average casting speed 1.7 m/min is shown in Figure 8. The cross-section size of the slab was 1530 mm × 250 mm and the numerical mesh was created from over 1.5 million nodes. Because the CC is a dynamic process and the casting speed varies in time, it is necessary to keep surface temperatures at constant values in order to achieve a high steel quality. Thus, optimization was carried out for two different casting speeds in order to obtain the optimal relationship between cooling intensities and the casting speed, see Table 3. These results can be used for the real CC process in case of steel grade S355 casting. The cooling intensity in loops 9 and 11 (last two loops at top surface) reaches its minimal allowed values for casting speed of 1.5 m/min. This means that for lower values of casting speed is even a better solution possibly exists. The last two cooling loops at top surface should therefore be replaced by smaller nozzles, which can operate smaller water flows (soft cooling). Moreover, significant savings of water consumption can be reached. In the case of casting speed 1.9 m/min, the maximal allowed cooling intensity values (physical limitations of pumps) were reached. This casting speed should not