Automatizace, počítačová simulace, výpočetní metody

Hutnické listy č.1/2010, roč. LXIII ISSN 0018-8069

Automatizace, počítačová simulace, výpočetní metody Automation Control, Computer Simulation , Computing Methods

automatizace, počítačová simulace, výpočetní metody Optimization of Technology and Control of a Slab Caster I. Off-line Numerical Model of Temperature Field of a Slab and Parametric Studies Optimalizace technologie a řízení bramového ZPO I. Off-line numerický model teplotního pole bramy a parametrické studie Doc. Ing. Josef Štětina, Ph.D., Prof. Ing. František Kavička, CSc., Vysoké učení technické v Brně, Fakulta strojního inženýrství, Prof. Ing. Jana Dobrovská, CSc., Vysoká škola báňská – Technická univerzita Ostrava, Fakulta metalurgie a materiálového inženýrství, Ing. Bohumil Sekanina, CSc., Ing. Tomáš Mauder, Vysoké učení technické v Brně , Fakulta strojního inženýrství

This paper brings new findings in the field of heat and mass transfer during solidification (crystallization) and cooling. The following thermo-physical phenomena and parameters are simulated: development of latent heat of phase or structural changes, prediction of the character of primary crystallization, speed of casting and its optimization, cooling of the mould, mould oscillation, intensity of cooling in the secondary zone by means of water or water-air nozzles, prediction of the position of the latest solidification. This original 3D numerical model is capable of simulating the temperature field of a caster. The presented model is a valuable computational tool and accurate simulator for investigating transient phenomena in caster operations, and for developing control methods, for the choice of an optimum cooling strategy to meet all the quality requirements, and for an assessment of the heat-energy content required for direct rolling. Článek přináší nové poznatky v oblasti přenosu tepla a hmoty v průběhu tuhnutí (krystalizace) a chlazení. Jsou simulovány následující termofyzikální pochody a parametry: vývin latentního tepla při fázových a strukturálních změnách, predikce charakteru primární krystalizace, rychlost lití a její optimalizace, chlazení krystalizátoru, oscilace krystalizátoru, intenzita chlazení v sekundární zoně vodními nebo vodo-vzdušnými tryskami, predikce místa posledního tuhnutí. Tento originální 3D numerický model umožňuje simulaci teplotního pole licího stroje. Předložený model je cenným výpočtovým prostředkem a přesným simulátorem pro výzkum přenosových jevů při práci licího stroje a pro vývoj řídících metod, výběr optimální strategie chlazení za účelem dosažení všech požadavků na kvalitu a stanovení obsahu tepelné energie potřebné pro přímé válcování. Byla provedena analýza vlivu licí rychlosti, chemického složení, přehřátí oceli a vlivu rozměrů bramy na výsledné teplotní pole bramy. K analýze je třeba vybrat takové výstupní veličiny, které lze jednoznačně stanovit a porovnávat, případně vybrat grafický průběh srovnávaného výstupního parametru. Jako nejvhodnější se jeví maximální metalurgická délka (délka tuhé fáze), maximální délka tekuté fáze, povrchová teplota v místě oblouku a povrchová teplota v místě před opuštěním klece sekundárního chlazení. Ukázalo se, že vliv chemického složení na délku tekuté i tuhé fáze je výrazný. Vliv licí rychlosti na teplotní pole byl sledován pro ocel 11325 v rozsahu 0,7 až 0,85 m/min, kdy průtok vody sekundárním chlazením je dle technologického předpisu zvyšován s licí rychlostí lineárně. Na grafech se ukázalo, že všechny sledované výstupní veličiny jsou na licí rychlosti závislé lineárně. Mírná odchylka této závislosti se projevila pro nižší rychlosti. Jako optimální rychlost lití byla stanovena 0,82 m/min. Tyto tzv. parametrické studie mohou sloužit k ověření používaných empirických vztahů, k sestavení technologických předpisů pro obsluhu ZPO, k provedení komplexní optimalizace a k nastavení dynamického modelu.

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1. Introduction Production of steels, alloys and metallurgical products in general is constantly developing. Materials with high utility parameters are more in demand and traditional production is being replaced by higher quality steel. More and more sophisticated aggregates using more sophisticated technological procedures are being implemented. In order to maintain competitiveness, diversify production and expand to other markets, it is necessary to monitor technological development. In the case of continuous casting, it is not possible to fulfil these requirements without the application of models of all caster processes dependent on thermalmechanical relationships. These models can be applied both off-line and on-line. An off-line model is the one where the calculation takes longer time than the time of the actual casting process. An on-line model runs in real time – taking the data directly from the operation – and its calculation takes the same time (if not less) than the actual process. These models will support the design of new and redesign of old machines, they will facilitate the identification and quantification of any potential defects and optimization of the various operational conditions in order to increase productivity and minimize the occurrence of defects. The process of the solidification of continuously cast steel is influenced by many factors and conditions, among which are the following: • Complete turbulent transient flow within a comprehensive geometry (input jet and liquid metal in the slab) • Thermo-dynamic reactions between the casting powder and the solidifying slab • Heat transfer between the liquid and solid powder on the surface of the slab • Dynamic movement of the liquid steel inside the mould on the liquid phase - mushy zone interface, including the influence of gravity, oscillations and the casting speed • Heat transfer in a super-heated melt considering turbulent flow • Transition (mixture) composition of the steel during the change of steel grade • Heat and mechanical interaction in the area of the meniscus between the solidifying meniscus, the solid powder and liquid steel • Heat transfer from the surface of the solidified shell into the space between the shell and the working surface of the mould (including the layers of the casting powder and the air gap) • Mass transfer of the powder during its vertical movement through the gap between the shell and the mould • Contact of the solidified slab with the mould and support rollers • Occurrence of crystals inside the melt • Process of micro-segregation and macro-segregation

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• Occurrence of shrinkages as a result of temperature contraction of the steel and the initialization of internal stress • Occurrence of stress and strain in the solidified shell as a result of external influences, such as friction inside the mould, bulging between the rollers, rolling, temperature stress and strain • Occurrence of cracks as a result of internal stress • Flow of steel as a result of electro-magnetic stirring and influence of the stirring on the temperature field and the primary structure • Occurrence of stress and strain as a result of unbending of the inside of the segments or in the rolling mill • Radiation between various surfaces • Cooling as a result of convection beneath the water or water-air jets With respect to the complexity of the investigation into the influence of all of the above-mentioned factors, it is not possible to develop a mathematical model that would cover all of them. It is best to group them according to the three main types of influence: • Heat and mass transfer • Mechanical • Structural The primary and deciding one is the influence of heat and mass transfer because it is the temperature field that gives rise to mechanical and structural influences. The development of a model of the temperature field (of a slab) with an interface for providing data for mechanical stress and strain models and structure models is therefore a top-priority task [1].

2. Model of the temperature field of a slab The 3D model had first been designed as an off-line version and later as an on-line version, so that it could work in real time. After correction and testing, it will be possible to implement it on any caster thanks to the universal nature of the code. The numerical model takes into account the temperature field of the entire slab (from the meniscus of the level of the melt in the mould to the cutting torch) using a 3D mesh containing more than a million nodal points. Solidification and cooling of a continuously cast slab is a global problem of 3D transient heat and mass transfer. If heat conduction within the heat transfer in this system is decisive, the process is described by the FourierKirchhoff equation. It describes the temperature field of the solidifying slab in all three of its states: at the temperatures above the liquidus (i.e. the melt), within the interval between the liquidus and solidus (i.e. in the mushy zone) and at the temperatures below the solidus (i.e. the solid state). In order to solve all this it is convenient to use the explicit numerical method of finite differences. Numerical simulation of the release of latent heats of phase or structural changes is carried out


by introducing of the enthalpy function dependent on temperature T, preferably in the form of an enthalpy related to the unit volume Hv. The latent heats are contained here. After the automated generation of the mesh (pre-processing) ties on the entry of the thermophysical material properties of the investigated system, including their dependence on temperature – in the form of tables or using polynomials. They are namely the heat conductivity k, the specific heat capacity c and density ρ of the cast metal.

1400 1200 1000

Enthalpy [kJ/m3]


The temperature distribution in the slabs described by the enthalpy balance equation 2

600 400 200 0 600

 ∂ T ∂ T ∂ T   ∂H ν ∂H ν ∂H ν ∂H ν   = k ⋅  2 + 2 + 2  +  u ⋅ + v⋅ + w⋅ ∂τ ∂ x ∂ y ∂z  ∂ x ∂ y ∂ z    2

800

800

1000

1200

Temperature [oC]

1400

1600

2

(1)

Fig. 1 Enthalpy function of typical carbon steel Obr.1 Závislost entalpie pro typickou uhlíkatou ocel

The simplified equation (1), suitable for application on radial-casters with great radius, where only the speed (of the movement of the slab) component w in the z-direction is considered, is:  ∂ 2T ∂ 2T ∂ 2T  ∂H ν ∂H = k ⋅  2 + 2 + 2  + w ⋅ ν ∂τ ∂y ∂z  ∂z  ∂x

(2)

Enthalpy Hv as a thermodynamic function of temperature must be known for each specific steel. It is dependent on the composition of the steel and on the rate of cooling. The dependence of the function H for typical carbon steel is shown in the Fig. 1. Fig. 2 shows that the task is symmetrical along the xaxis, it is therefore sufficient to investigate only half of the cross-section with the following boundary conditions (3a-3e)

1. T = Tcast at the meniscus 2.

−k

∂T =0 ∂n

at the plane of symmetry

(3a) (3b)

3. − k ∂T = h.(T in the mould surface − Ta )

(3c)

4 4 4. − k ∂T = h ⋅ (T surface − Ta ) + σ ⋅ ε ⋅ (Tsurface − Ta )

(3d)

∂n

∂n

Fig. 2 The mesh and definition of the coordinate system Obr.2 Výpočtová síť a definice souřadného systému

The initial condition for solving is the setting of the initial temperature in individual points of the mesh. A suitable value is the highest possible temperature, i.e. the casting temperature. Fig. 3 illustrates the thermal balance of an elementary volume (general nodal point i, j, k) of the network.

in the secondary and tertiary cooling zone 5.

−k

∂T =q ∂n

beneath the support rollers

(3e)

Fig. 3

The thermal balance diagram of the general nodal point of the network Obr.3 Diagram teplotní bilance obecného uzlu výpočtové sítě

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Automatizace, počítačová simulace, výpočetní metody Automation Control, Computer Simulation , Computing Methods An unknown enthalpy of the general nodal point of the slab in the next time step (τ+∆τ) is expressed by the explicit formula: H v(i,τ +j,k∆τ ) = H v(i,τ )j,k + (QZ1i, j + QZ i, j + QY1i, j + QYi, j + QX1 + QX )

∆τ ∆x ⋅ ∆y ⋅ ∆z

(4) The heat flow through the general nodal point (i, j, k) in the z-direction is described by the following equations

(

)

QZ i, j = VZ i, j,k Ti,(j,τ k) + 1 − Ti,τj,k − A z ⋅ w ⋅ H (vτi,)j,k

where

(5)

VZi,j,k = k . Az / ∆z

Fig. 1 indicates how the temperature model for the calculated enthalpy in the equation (4) determines the unknown temperature. The enthalpy function is not known as an analytical function but as a set of table values, and therefore the inverse calculation of the temperature is a very demanding problem numerically. In the dynamic model where the calculation must run at least as fast as is the flow of the process in real time, the method, in which the interpolation values are calculated at 0.1 °C intervals even before the actual calculation, was chosen. The temperature for the relevant enthalpy is then determined using modern search methods.

3. Parametric study of slab casting The original off-line model of the temperature field and its calculation speed makes it possible to perform the so-called parametric studies, i.e. the analyses of the effect of individual input technological parameters and properties on the resultant temperature field. The results of these parametric studies could serve for the verification of the empirical relationships used, for establishing technological procedures for the caster operators, for the carrying out of comprehensive optimization and for the setting up of the dynamic model. Regarding the fact that the result of the modelling process is a 3D temperature field, it is necessary to select such output parameters that can be uniquely defined and compared, or to select the graphic output of the compared output parameter in order to assess the effect of the chemical composition. The most suitable for the comparison seems to be the maximum metallurgical length, the maximum length of the liquid phase, the surface temperature at the unbending point and the surface temperature at the point just before leaving the caster cage. The courses of the temperatures of the same points of the cross-section were selected as the basic graphical representation for comparison along the entire length of the caster, in combination with the graph showing the increase in the thickness of the shell.

46


There are three types of graphs drawn in order to illustrate the results as clearly and as completely as possible. Fig. 5 shows the temperature history of six points of the cross-section of the slab (i.e. in the centre of the slab, in its corners and in the middle of the sides of the surface) in the way that the cross-section passes through the caster (from the meniscus of the steel in the mould all the way down to the cutting torch). The distance from the meniscus inside the mould is plotted on the horizontal axis. The width of the horizontal yellow strip illustrates the temperature interval for the relevant class of steel. The width of the vertical yellow strip illustrates the distance between the iso-liquidus and iso-solidus (i.e. the width of the mushy zone) in its maximum values. Furthermore, the graph indicates two surface temperatures where the pyrometers were positioned. The red vertical dashed lines are the boundaries between the individual segments, and the blue vertical lines represent the meniscus of the steel inside the mould, the bottom edge of the mould, the unbending point and the end of the caster cage. The next graph illustrates the temperature field in the diagram of the caster, where the shades of blue represent solidified steel, the shades of yellow represent liquid steel and the red represents the mushy zone. The last graph is the course of the iso-liquidus and iso-solidus unrolled along the longitudinal section through the entire continuous casting. This picture gives a clear idea of the shape of the mushy zone, which closely corresponds to the structure and any potential internal defects. 3.1 The effect of the chemical composition on the resultant temperature field A real operation of continuous casting casts up to several hundred grades of steel. It would therefore be difficult to set the continuous casting and other relevant technological parameters for all of them. That is why steels are subdivided into groups, mostly according to their carbon content, preferably according to the socalled equivalent carbon content, given by: C eq = C − 0.1 ⋅ Si + 0.04 ⋅ Mn − 0.04 ⋅ Cr + 0.1 ⋅ Ni − 0.1 ⋅ Mo

(6)

A single grade of steel was selected from each group for the analyses below. Table 1 contains the recommended compositions of these steels, together with the temperatures of the liquidus and solidus. Fig. 4 illustrates an example of the dependence of the thermophysical properties on the temperature for the 11325 grade [2]. Fig. 5 presents the calculated temperature field for this grade of steel. These calculations were performed also for the remaining grades [3].



8000

900

50

7600

40

7200

30

6800

20

6400

800

700

600

500

10 600

800

1000

1200

Density [kg/m3]

60

Heat conductivity [W/m.K]

Specific heat capacity [J/kg.K]

Steel 11325 1000

6000 1600

1400

Temperature [oC] Heat conductivity

Specific heat capacity

Density

Fig. 4 Thermo-physical properties of the steel grade 11325 steel from the group 2 (first row of the Table 1.) Obr.4 Tepelné vlastnosti oceli třídy 11325 ze skupiny 2 (první řada Tab.1) Tab. 1 Selected grades of steel with their compositions used for calculation Tab. 1 Vybrané třídy ocelí a jejich složení použité pro výpočet

Class Group Ceq

C

Mn

Si

P

S

Cu

Ni

Cr

Mo

V

Ti

Al

Nb

Tsol

Tliq

11325

2

0.067 0.050 0.225 0.025 0.010 0.010 0.150 0.150 0.150 0.040 0.050 0.0025 0.045 0.030 1499.8 1529.8

21026

5

0.235 0.150 1.075 0.300 0.0175 0.010 0.150 0.200 0.100 0.040 0.045 0.001 0.040 0.015 1451.4 1514.2

31087

3

0.275 0.190 1.450 0.200 0.015 0.010 0.100 0.150 0.100 0.040 0.010 0.001 0.040 0.030 1438.7 1510.6

11500

4

0.326 0.270 0.550 0.275 0.015 0.010 0.150 0.150 0.125 0.040 0.050 0.025 0.040 0.030 1423.2 1507.4

13180

6

0.826 0.75 1.050 0.250 0.0175 0.010 0.125 0.200 0.150 0.050 0.100 0.050 0.040 0.025 1322.7 1467.7

In order to analyse the influence of the chemical composition on the temperature field more clearly, the other parameters of continuous casting were selected identically, i.e. the casting speed 0.8 m/min, the superheating temperature 30 °C and the profile of the slab 1530×250 mm, just like the flow of water through the secondary-cooling zone. In practice, a different cooling mode is selected for each different grade of steel.

Fig. 5 Temperature field of the slab made of steel grade 11325 Obr.5 Teplotní pole bramy z oceli třídy 11325

Figures 6 and 7 prove that the effect of the chemical composition on the resultant temperature field evaluated by the above-mentioned output parameters is significant.

47

Automatizace, počítačová simulace, výpočetní metody Automation Control, Computer Simulation , Computing Methods Steel Group 2 3 4

1


Tab. 2 Effect of casting speed Tab. 2 Vliv rychlosti lití

5

Length of concasting [m]

24

Casting speed m/min 0.7 0.75 0.8 0.85

20 16 12

Solid Mushy Liquid

8 4 0

Fig. 6

Comparison of the length of the liquid phase and the metallurgical length for various grades of steel Obr.6 Srovnání délek tekuté fáze a metalurgických délek různých tříd oceli

Metallurgical length Length of liquid phase Temperature in unbending part Temperature Metallurgical length in end of cage

2424

LLIQ

LMET

M 15.717 16.346 17.611 18.915

m 16.849 18.694 20.157 21.622

1000

1000

920

1818

880

1616

840

1414

800

Temperature [°C]

920

880

Temperature [oC]

Metallurgical length [m]

Length [m]

960

960

2020

840

800

0

0

0.2

0.2

0.4 Ceq [wt %]

0.4

0.6

0.6

a) Casting speed 0.70 m/min

0.8

0.8

Ceq [wt %]

Fig. 7

The effect of the chemical composition on the resultant parameters Obr.7 Vliv chemického složení na výsledné parametry

3.2 Effect of the casting speed The casting speed is a basic technological parameter. In calculations, the results of which are represented here (for the steel grade 11325 steel), the operating range of the speed is considered to be between 0.7 to 0.85 m/min. The flow of water through the secondary-cooling zone, according to the technological regulations, increases linearly. The other input parameters are again left constant, i.e. especially the superheating temperature of 30 °C. Higher speeds need not be investigated because the metallurgical length exceeds the length of the cage, which is unacceptable. On the other hand, lower casting speeds are used only for a short-term, e.g. in the case that there is the risk of breakout or when the tundish is being exchanged.

b) Casting speed 0.75 m/min

c) Casting speed 0.80 m/min

48

Tend °C 871 892 906 920

The graphs in Figures 8 and 9 and Table 2 show that there is a linear dependence of all of the monitored parameters (the metallurgical length, the length of the liquid phase, the temperature of the unbending part and the temperature at the end of the cage) on the casting speed. A slight deviation can be seen for lower speeds, which corresponds with the setting of the secondary cooling. A casting speed of 0.82 m/min seems to be the optimum for all of these parameters.

Length of liquid phase Temperature in unbending part Temperature in end of cage

2222

Tunbendin g °C 932 954 972 990



Metallurgical length Length of liquid phase Temperature in unbending part Temperature in end of cage

22

1000

960 20

Fig. 8 Comparison of the temperature fields for various casting speeds, 11325 steel, superheating 30 °C Obr.8 Srovnání teplotních polí pro různé rychlosti lití, ocel 11325, přehřátí 30oC

18

880

840

Metallurgical length Length of liquid phase Temperature in inbending part Temperature in end of cage

16

Temperature [oC]

d) Casting speed 0.85 m/min

Length [m]

920

800 10


22

1000

880

840

800 0.72

0.76

0.8

50

Tab. 3 Vliv teploty přehřátí Tab. 3 Influence of the superheating temperature Temperature [oC]

Length [m]

920


40

Obr.10 Vliv teploty přehřátí, ocel třídy 11325, rychlost lití 0.8 m/min Fig. 10 Influence of the superheating temperature, steel grade 11325 class steel, speed 0.8 m/min

960

16

30

Superheating temperature [oC]

20

18

20

Tsuperheatin g °C 50 40 30 20 10

LLIQ m 19.126 18.780 18.388 18.057 17.695

LMET m 20.469 20.121 19.710 19.373 18.999

Tunbending °C 984 977 972 967 963

Tend °C 914 912 906 904 900

0.84

Casting speed [m/min] Effect of the casting speed on the selected parameters, steel grade 11325, superheating 30 °C Obr. 9 Vliv rychlosti lití na vybrané parametry, ocel 11325, přehřátí 30oC

3.4 Influence of the slab width

Fig. 9

3.3 Influence of the superheating temperature The casting temperature must always be higher than thetemperature of the liquidus, in order to ensure sufficient transport of liquid steel through the casting nozzle. From the operation point of view, it is desirable to increase the casting speed when the superheating temperature is lower in order to empty the tundish. Fig. 10 and Table 3 summarise information on the input parameter. They show how the level of the superheating of the melt above the temperature of the liquidus influences the metallurgical length. The lower the superheating, the shorter the metallurgical length – this enables the introduction of a higher casting speed. This finding is in accordance with the requirement for ensuring timely emptying of the tundish. Calculations proved that the effect of the superheating on the surface temperatures is limited.

The design of the mould of the slab caster usually makes it possible to alter the width of the slab. At operation at the EVRAZ VÍTKOVICE STEEL, it is possible to alter the width of the slab within the range from 800 to 1600 mm. Fig. 11 shows the graphics output from the calculation of the temperature field of the slab for a width within the range from 1200 to 1600 mm. It is necessary to state that a change in the width does not bring about any changes in the configuration of the secondary cooling, which leads to under-cooling of the corners. The graphs also show a top-view history of the iso-liquidus and iso-solidus. It appears that the influence of the width on the metallurgical length is smaller than expected. Considerably greater influence effects the change of the blue iso-solidus, i.e. the shape of the (double) cone. With a marginal slab width of 1600 mm, its edges are out of reach of the cooling jets. This manifests itself in the form of a peak on the iso-solidus and iso-liquidus curve along the edge of the slab (Fig. 11a), which could bring about the occurrence of internal defects. Fig. 12 shows that, from this point of view, the most frequently used width of 1530 mm is the maximum allowable. With the slabs of the smaller widths, the course of the

49



iso-lines is favourable and their more frequent casting would bring about a reduction in the production of continuously cast steel.

d) Width 1300 mm

a) Width 1600 mm

e) Width 1200 mm

b) Width 1500 m

Fig. 11 Temperature field for slab widths 1600 to 1200 mm, steel grade 11325, speed 0.8 m/min Obr. 11 Teplotní pole bramy o tloušťkách 1600 až 1200 mm. ocel třídy 11325, rychlost lití 0.8 m/min


22

1000

960 20

18

Length [m]

880

Metallurgical length Length of liquid phase Temperature in unbendig part Temperature in end of cage

16

840

Temperature [oC]

920

800

c) Width 1400 mm 1200

1300

1400

1500

1600

With of slab [mm]

Fig. 12 Effect of the width of the slab, steel grade 11325, speed 0.8 m/min

50



Obr. 12 Vliv tloušťky bramy, ocel třídy 11325, rychlost lití 0.8 m/s Tab. 4 The influence of the slab width Tab. 4 Vliv tloušťky bramy

Width mm 1600 1500 1400 1300 1200

LLIQ m 19.037 18.151 18.023 17.921 17.775

LMET m 20.389 19.501 19.388 19.286 19.137

Tunbending °C 976 972 969 965 963

Tend °C 916 901 899 898 897

b) Thickness 180 mm

3.5 Influence of the slab thickness On some machines it is possible to alter the thickness of the slab. The mould of the caster in question can be set to three different slab thicknesses i.e. 250, 180 and 145 mm (Fig. 13). In this study, the calculations have been carried out for the same grade of steel and superheating temperature. However, the casting speed and cooling mode are set to the usual values for each profile because there is no sense in calculations using the same speed and cooling.

c) Thickness 250 mm Fig. 13 Temperature field for slab thickness 145 to 250 Obr.13 Teplotní pole bramy o tlouťkách 145 až 250

Tab. 5 The influence of the slab thickness Tab. 5 Vliv tloušťky bramy

Thickness mm 145 180 250

a)

Thickness 145 mm

Width mm 1530 1530 1530

LLIQ m 14.114 19.094 17.611

LMET m 19.967 20.471 20.157

Tunbending °C 1111 1106 972

Tend °C 916 901 906

4. Conclusions This paper introduces a 3D numerical model of the temperature field (for continuous casting of steel) in the form of an in-house software. The model includes the main thermo-dynamic transfer phenomena during the solidification of continuous casting. The temperature model is used for monitoring and for parametric studies of the changes in chemical composition, the casting speed, the superheating temperature and the dimensions of the continuously cast slab. Acknowledgments This analysis was conducted using a program devised within the framework of the projects GA CR No. 106/08/0606, 106/08/1243 106/09/0940 and the MPO CR project No. FT-TA4/048.

51



Literature

Nomenclature c specific heat capacity [J/kg.K] Ceq equivalent carbon content [wt. %] C mass composition of carbon [wt. %] h heat transfer coefficient [W/m2.K] Hv volume enthalphy [J/m3] k heat conductivity [W/m.K] LLIQ length of the liquid phase [m] LMET metallurgical length [m] T temperature [K] Ta ambient temperature [K] Tcast melt temperature [oC] Tunbending temperature at the unbending part [oC] Tend temperature at the end of cage [oC] q specific heat flow [W/m2] x,y,z axes in the given direction [m] u,v,w casting speed in the given direction [m/s] ρ density [kg/m3] σ Stefan-Bolzmann constant [W/m2.K4] ε emissivity [-] τ time [s]

[1] Brimacombe, J. K., (1999), “The Challenge of Quality in Continuous Casting Process”. Metallurgical and Materials Trans, B, Volume 30B, pp. 553-566. [2] Miettinen, J. and Louhenkilpi, and Laine, J., (1996), Solidification analysis package IDS. Proceeding of General COST 512 Workshop on Modelling in Materials Science and Processing, M.Rappaz and M. Kedro eds., ECSC-EC-EAEC, Brussels, Luxembourg. [3] Richard, A. and Harding, Kai Liu and Beckermann, Ch, (2003), A transient simulation and dynamic spray cooling control model for continuous steel casting, Metallurgical and materials transactions, volume 34B, pp 297-302. [4] Stetina, J. (2007), “The dynamic model of the temperature field of concast slab”. Ph.D. Thesis, Technical University of Ostrava, Czech Republic. [5] Thomas, B. G. and O’Malley, R. J. and Stone, D. T., (1998), Measurement of temperature, solidification, and microstructure in a continuous cast thin slab. Paper presented at Modelling of Casting, Welding and Advanced Solidification Processes VIII, San Diego, CA, TMS.

Recenze: Prof. Ing. Milan Vrožina, CSc. Doc. Dr. Ing. René Pyszko

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2 THETA ASE, s.r.o. Český Těšín, CZ Instytut Metalurgii Żelaza w Gliwicach, PL Komisja Śladowej Analizy Nieorganicznej PAN, PL VILLA LABECO, s.r.o. Spišská Nová Ves, SK

Vás srdečně zvou na 8. ročník česko - polsko – slovenské konference

Hutní a průmyslová analytika 2010 19. - 23. 4. 2010 Valtice Organizační zabezpečení : 2 THETA ASE, s..r.o. - Dagmar Krucinová, Ing. Václav Helán P.S. 103, 737 01 Český Těšín Tel/fax: +420 558 732 122, mobil + 420 602 240 553 [email protected] Instytut Metalurgii Żelaza, Dr. Grażyna Stankiewicz VILLA LABECO, s.r.o., RNDr. Marian Kovaľ

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Automatizace, počítačová simulace, výpočetní metody

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