EP2279053B1 - Method for the continuous casting of a metal strand - Google Patents

Description

The present invention relates to a method for continuously casting a metal strand.

Specifically, the invention relates to a method for continuously casting a metal strand, in particular a steel strand, in a continuous casting, wherein a strand with a trapped by a strand shell, liquid core drawn from a cooled continuous casting mold, supported in a continuous casting mold downstream strand support means and cooled with coolant , where thermodynamic state changes of the entire strand in a mathematical simulation model, taking into account the physical parameters of the metal, the thickness of the strand and the constantly measured extraction speed are also calculated.

From the DE 4417808 A1 For example, there is known a method for continuously casting a metal strand in which a strand having a liquid core enclosed by a strand shell is drawn out of a cooled mold, then supported in a strand support and cooled with coolant. The state changes occurring in the course of the continuous casting process are also calculated in real time for the entire strand by means of a mathematical simulation model, including the two-dimensional heat equation, and the cooling of the strand is set as a function of the calculated thermodynamic state changes.

Due to the two-dimensionality of the heat equation used, it has not been possible to calculate the heat conduction and the associated state changes in all directions (the strand thickness, the strand width and the strand extraction direction) of the metal strand and adjust the temperature profile depending on the calculated state changes by means of strand cooling targeted. In addition, there were differences between the calculated and the actual thermodynamic effects, which were not taken into account in the simulation model By freezing point.

From the EP 0 997 203 A1 a method for controlling a cooling section is known, wherein by means of a mathematical process model, the temperature profile over the length of the tape is calculated. Subsequently, the cooling section is controlled taking into account a Referenzternperaturverlaufs and the calculated temperature profile.

The object of the invention is to provide a method of the type mentioned, with which the accuracy of the simulation of the thermodynamic changes in state of the entire strand can be further increased and in connection with the cooling of the product quality of the metal strand and the production efficiency of the continuous casting process can be improved.

This object is achieved by a method of the type mentioned, in which a three-dimensional heat equation is solved numerically in real time in the mathematical simulation model and the cooling of the strand is set taking into account the calculated state changes.

By means of the method from the DE 4417808 A1 It is possible to calculate thermodynamic state changes as a function of the two-dimensional heat conduction in real time and to influence the temperature profile of the strand by means of strand cooling. According to the method of the invention, it is possible thermodynamic state changes by means of a non-linear, transient heat equation as a function of the three-dimensional heat conduction, namely in strand thickness direction, in the strand width direction and in the strand longitudinal direction, ie. in the extension direction of the strand to calculate in real time and to influence by means of strand cooling targeted. As a result, the thermodynamic changes in state can be calculated with greater accuracy and be specifically influenced by means of a coordinated strand cooling. In the mathematical simulation model, the strand is divided into individual volume elements, ie so-called discretized, each discrete Volume element has an extension in strand longitudinal direction, in the strand thickness direction and in the strand width direction. By means of this discretization individual nozzles of the strand cooling can be assigned to one or more discrete volume elements of the strand and thereby firstly the thermodynamic state changes in these volume elements taking into account the heat conduction in all spatial dimensions and the determined by the strand cooling heat quantity can be determined with high accuracy, and secondly, by means of these nozzles, the thermodynamic properties of the strand can be influenced very targeted and with high efficiency.

In a particularly advantageous embodiment, the three-dimensional heat equation is solved numerically in the mathematical simulation model of the method according to the invention taking into account the temperature-dependent change in density of the metal strand. It is known to the person skilled in the art that the change in density of metal as a function of the temperature can assume significant proportions. For example, in the continuous casting process, the density of steel increases from approx. 7000 kg / m ³ at 1550 ° C (melt temperature in the casting distributor) to approx. 7800 kg / m ³ at 300 ° C (solidified strand). The density changes are also relevant in the continuous casting process in connection with the heat conduction equation in the determination of the solidification point. As Durcherstarrungspunkt that point in strand extraction direction is referred to, from which the metal strand is completely solidified, ie. the metal strand no longer has a liquid core. The most accurate calculation of the solidification point is extremely advantageous in any case. If the location of the solidification point underestimated, ie. If the calculated point is less far removed from the mold than the actual point, this can lead to very dangerous casting situations (eg strand breakage). On the other hand, the allowable casting speed is unnecessarily limited in overestimating the through-solidification point, which in turn would degrade the productivity of the equipment.

A further, particularly advantageous embodiment of the method according to the invention can be achieved if approximated equations for the enthalpy, which have the exact mass and the exact enthalpy for the entire strand, are used in the numerical solution of the heat equation, taking temperature-dependent density changes of the metal strand into account. It should be noted at this point that the exact three-dimensional, nonlinear and unsteady heat equation with regard to the temperature-dependent density change is still unsolved. The thermal equation used today without taking into account the temperature-dependent density change are only rough approximations of the exact equation and their solutions may differ significantly from the exact solution. By using approximate equations for the enthalpy with global - ie. however, if the entire strand is considered - the exact mass and the exact enthalpy - it is ensured that these essential thermodynamic state variables correspond to the exact values.

The method according to the invention can be carried out particularly favorably if either a finite volume method or a finite element method is used to solve the heat equation in the mathematical simulation model. The heat equation is a parabolic partial differential equation that can be solved using standard methods of numerical mathematics, in particular the finite volume method or finite element method (see Chapter 19: Numerical Mathematics by IN Bronstein, KA Semendjajew, G. Musiol, H. Mühlig: Paperback of Mathematics, Publisher Harri Deutsch, 6th Edition, 2005 ).

In a particularly favorable manner, the method according to the invention is carried out when the thermodynamic state changes due to the spatial symmetry are calculated only for a quarter of the strand cross-section. This simplification can be made without loss of accuracy due to the spatial symmetry of the strand cross-section and the time-varying boundary conditions and enables the three-dimensional heat equation can be solved with high accuracy even by relatively low-performance process computers.

The method according to the invention can be used without restrictions when casting metal strands with billet, billet, slab or thin slab cross section Any dimensions can be used to improve the quality of the cast metal strands.

• Fig. 1 eine Stranggussanlage in schematischer Seitenansicht
• Fig. 2 eine schematische Darstellung des diskretisierten Metallstrangs
• Fig. 3 ein Vergleich von Lösungen unterschiedlicher Formulierungen von Wärmeleitungsgleichungen

Further advantages and features of the present invention will become apparent from the following description of non-limiting embodiments, reference being made to the following figures, which show the following:

• Fig. 1 a continuous casting plant in a schematic side view
• Fig. 2 a schematic representation of the discretized metal strand
• Fig. 3 a comparison of solutions of different formulations of heat equation

A cooled mold 1 is fed with liquid steel 2, which is supplied from an intermediate vessel 3. The forming in the mold 1, a liquid core 4 and initially only a thin strand shell 5 having, strand 6 is an arcuate strand support means 7, which is provided with support rollers 8 and supports the strand at the top and at the bottom, redirected to the horizontal where, after solidification, it is either cut up or transported further as a continuous strand. For the cooling of the strand 6 7 coolant-supplying nozzles 10 are provided along the strand support means, of which in the drawing only those are drawn on the strand top at the beginning of the strand support means 7. In this case, one or more nozzles 10 are connected to a respective supply line 11. The amount of coolant applied by the nozzles to the strand can be varied by means of a continuously adjustable valve 12, to which a flow measuring device 13 is arranged downstream. Each valve 12 is adjustable via an actuator 14 that can be actuated via a control element 16 controlled by a central process computer 15. Each flow measuring device is coupled via an input unit 17 to the process computer 15, which in turn drives all control elements 16 via an output unit 18. In the input unit 17 of the process computer 15, for example, the physical parameters of the metal to be cast, in the present case of the steel 2, namely the temperature-dependent values of the density, the specific Heat capacity and thermal conductivity, further the flow-dependent spray pattern of the location-dependent arranged nozzles 10, the location-dependent role division 9, the optionally location-dependent strand thickness, the strand width and the continuously measured casting speed of the continuous casting plant are entered.

According to the invention, the strand 6 is cooled in a controlled manner at specific, either fixed or variable, positions of the strand support device 7. The control of the strand cooling takes place taking into account the thermodynamic state changes of the entire strand 6 by the release in real time of a three-dimensional heat equation using the process computer 15th

wobei
t  Zeit in [s]
x  die Koordinate in Strangdickenrichtung in [m]
y  die Koordinate in Strangbreitenrichtung in [m]
z  die Koordinate in Auszugsrichtung bzw. der Stranglängsachse in [m]
$\frac{\partial }{\partial t}$

  partielle Ableitung nach der Zeit t
$\frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}$

  partielle Ableitungen nach dem Ort x, y, z
x   Ortsvektor in einem rechtwinkeligen Koordinatensystem in [m]
ρ  Dichte in [kg/m³]
E_mass ( x ,t)  Massenbezogene Enthalpie an der Stelle x zur Zeit t in [J/kg]
ξ  Dimensionslose Laufvariable $$u\left(\overrightarrow{x},t\right)=\underset{T_{\mathit{ref}}}{\overset{T\left(\overrightarrow{x},t\right)}{\int }}\lambda \left(\xi \right)\cdot d\xi$$

v_cast (t)  Auszugsgeschwindigkeit des Strangs zur Zeit t in [m/s]For example, a three-dimensional, non-linear and transient heat equation in an enthalpy formulation is $$\rho \left(\frac{\partial e_{\mathit{measure}}\left(\overrightarrow{x},t\right)}{\partial t}+v_{\mathit{cast}}\left(t\right)\frac{\partial e_{\mathit{measure}}\left(\overrightarrow{x},t\right)}{\partial z}\right)=\frac{{\partial }^{2}u\left(\overrightarrow{x},t\right)}{\partial x^2}+\frac{{\partial }^{2}u\left(\overrightarrow{x},t\right)}{\partial y^2}+\frac{{\partial }^{2}u\left(\overrightarrow{x},t\right)}{\partial z^2}$$

in which
t time in [s]
x the coordinate in strand thickness direction in [m]
y is the coordinate in the strand width direction in [m]
z the coordinate in the extension direction or the longitudinal axis of the strand in [m]
$\frac{\partial }{\partial t}$

partial derivative after time t
$\frac{\partial }{\partial x}.\frac{\partial }{\partial y}.\frac{\partial }{\partial z}$

partial derivatives according to the location x, y, z
x Position vector in a rectangular coordinate system in [m]
ρ density in [kg / m ³ ]
E _mass ( x , t ) mass-related enthalpy at point x at time t in [J / kg]
ξ Dimensionless run variable $$u\left(\overrightarrow{x},t\right)=\underset{T_{\mathit{ref}}}{\overset{T\left(\overrightarrow{x},t\right)}{\int }}\lambda \left(\xi \right)\cdot d\xi$$

v _cast ( t ) Pull rate of the strand at time t in [m / s]

In this heat equation, temperature-dependent changes in density of the strand 6 are disregarded. Since the density of steel 2 increases from 7000 kg / m ³ at 1550 ° C. to 7800 kg / m ³ at 300 ° C., this simplification leads to inaccuracies in the calculation of the thermodynamic state changes. It has been found that in this heat equation, the Durcherstarrungspunkt is underestimated, ie. that the actual through-solidification point is farther from the mold 1 than the calculated through-solidification point. In order to avoid disadvantageous and possibly even dangerous casting situations, it is necessary to use a density value p in the region of the maximum density of the steel 2, which subsequently results in the max. permissible casting speed significantly reduced.

wobei

T( x ,t)

Temperatur an der Stelle x zur Zeit t in [° K]

ρ(T( x ,t))

Dichte des Metallstrangs bei der Temperatur T in [kg/m³]

A second formulation of a nonlinear, three-dimensional and transient heat equation is $$\rho \left(T\left(\overrightarrow{x},t\right)\right)\left(\frac{\partial e_{\mathit{measure}}\left(\overrightarrow{x},t\right)}{\partial t}+v_{\mathit{cast}}\left(t\right)\frac{\partial e_{\mathit{measure}}\left(\overrightarrow{x},t\right)}{\partial z}\right)=\frac{{\partial }^{2}u\left(\overrightarrow{x},t\right)}{\partial x^2}+\frac{{\partial }^{2}u\left(\overrightarrow{x},t\right)}{\partial y^2}+\frac{{\partial }^{2}u\left(\overrightarrow{x},t\right)}{\partial z^2}$$

in which

T ( x , t )

Temperature at point x at time t in [° K]

ρ ( T ( x , t ))

Density of the metal strand at the temperature T in [kg / m ³ ]

This formulation of the heat equation is global, ie when the entire strand is considered, true to mass, but incorrect in terms of enthalpy. It has been shown that in this heat equation, the through-solidification point is overestimated, ie. that the actual through-solidification point is less far from the mold than the calculated through-solidification point. Thus, the use of this equation is indeed problematic with respect to any adverse casting situations, however, the max. allowable casting speed unnecessarily limited, resulting in reduced productivity of the plant.

wobei

E_trans ( x,t)

Transformierte massenbezogene Enthalpie an der Stelle x zur Zeit t

Advantageously, the heat equation is used $$\left(\frac{\partial e_{\mathit{trans}}\left(\overrightarrow{x},t\right)}{\partial t}+v_{\mathit{cast}}\left(t\right)\frac{\partial e_{\mathit{trans}}\left(\overrightarrow{x},t\right)}{\partial z}\right)=+\frac{{\partial }^{2}u\left(\overrightarrow{x},t\right)}{\partial x^2}+\frac{{\partial }^{2}u\left(\overrightarrow{x},t\right)}{\partial y^2}+\frac{{\partial }^{2}u\left(\overrightarrow{x},t\right)}{\partial z^2}$$

in which

E _trans ( x , t )

Transformed mass-enthalpy at point x at time t

Two approaches for a, with respect to the mass and the enthalpy, globally correct, transformed enthalpy E _trans ( x , t ) used. The thermodynamic conditions in the strand 6 change significantly at the through-solidification point 19, since the strand 6, viewed in the casting direction, above the through-solidification point has a liquid core 4, which communicates with the liquid steel 2 of the mold 1. The ferrostatic pressure in this area presses the already solidified strand shell 5 against the rollers 8 of the strand support device 7, whereby the strand shrinkage due to the temperature-dependent change in density of the steel 2 is compensated by inflowing, liquid steel 2 in this area. Below the Durcherstarrungspunkts 19 such compensation does not take place.

Hingegen verwendet man unterhalb des Durcherstarrungspunkts 19 folgenden Ansatz $${\mathrm{E}}_{\mathrm{trans}}\left(\overrightarrow{x},t\right)=\underset{T_{\mathit{ref}}}{\overset{T\left(\overrightarrow{x},t\right)}{\int }}\rho \left(\xi \right)\cdot {\dot{\mathit{E}}}_{\mathit{mass}}\left(\xi \right)\cdot d\xi$$

Hierin bedeuten

T_ref

eine beliebige, aber konstante Referenztemperatur (üblicherweise 25 °C)

T_tund

Temperatur des Metalls im Gießspiegel in [° K]

Ė_mass ( x ,t)

Zeitliche Ableitung der massenbezogenen Enthalpie

Above the Durcherstarrungspunkts 19 is the approach $${\mathrm{e}}_{\mathrm{trans}}\left(\overrightarrow{x},t\right)=\underset{T_{\mathit{ref}}}{\overset{T\left(\overrightarrow{x},t\right)}{\int }}\left[\rho \left(\xi \right)\cdot {\dot{\mathit{e}}}_{\mathit{measure}}\left(\xi \right)-\dot{\rho }\left(\xi \right)\left(e_{\mathit{measure}}\left(T_{\mathit{tund}}\right)-e_{\mathit{measure}}\left(\xi \right)\right)\right]d\xi \mathrm{,}$$

On the other hand, below the solidification point 19, the following approach is used $${\mathrm{e}}_{\mathrm{trans}}\left(\overrightarrow{x},t\right)=\underset{T_{\mathit{ref}}}{\overset{T\left(\overrightarrow{x},t\right)}{\int }}\rho \left(\xi \right)\cdot {\dot{\mathit{e}}}_{\mathit{measure}}\left(\xi \right)\cdot d\xi$$

Herein mean

T _ref

any but constant reference temperature (usually 25 ° C)

T _tund

Temperature of the metal in the pouring mirror in [° K]

Ė _mass ( x , t )

Time derivative of mass enthalpy

$${\overrightarrow{x}}_{\mathit{Lag}}\left(t\right)=\left(\begin{array}{c}x\\ y\\ \mathit{CastLg}\left(t\right)-z\end{array}\right)$$

wobei

t_start

Zeitpunkt der Entstehung des diskreten Volumenelements in der Kokille in [s]

The heat equation becomes Lagrange coordinates x _Lag transformed, ie viewed by a moving along with the strand extraction movement observer. The transformation is $$\mathit{CastLg}\left(t\right)={\int }_{t_{\mathit{begin}}}^t{v}_{\mathit{cast}}\left(\zeta \right)\cdot d\zeta$$

$${\overrightarrow{x}}_{\mathit{lag}}\left(t\right)=\left(\begin{array}{c}x\\ y\\ \mathit{CastLg}\left(t\right)-z\end{array}\right)$$

in which

t _start

Time of formation of the discrete volume element in the mold in [s]

The heat equation in Lagrange coordinates is then $$\frac{\partial e_{\mathit{trans}}\left({\overrightarrow{x}}_{\mathit{lag}}\left(t\right)\right)}{\partial t}=\frac{{\partial }^{2}u\left({\overrightarrow{x}}_{\mathit{lag}}\left(t\right)\right)}{\partial x^2}+\frac{{\partial }^{2}u\left({\overrightarrow{x}}_{\mathit{lag}}\left(t\right)\right)}{\partial y^2}+\frac{{\partial }^{2}u\left({\overrightarrow{x}}_{\mathit{lag}}\left(t\right)\right)}{\partial z^2}$$

This heat equation is solved by the process computer 15 in real time by means of the finite volume method. This standard method of numerical mathematics is known to the person skilled in the art and works with discrete volume elements of the strand 6. For each volume element 20, therefore, the simple three-dimensional heat equation described in the v- _cast moving, element-fixed coordinate system is to be solved. This is performed periodically for a plurality of volume elements 20, resulting in the time-varying temperature field of the entire strand 6. Out Fig. 2 it can be seen that the strand 6 is divided into discrete volume elements 19, for example, 10 cm edge length. The volume elements 19 are produced in the mold and tracked in accordance with the casting speed through the continuous casting plant.

As in Fig. 2 As a result of this spatial symmetry in the strand width and strand thickness direction, it is advantageous to calculate the thermodynamic state changes only in one quadrant 20, ie one quarter, of the strand cross-section.

To solve the heat equation, however, the initial and - depending on the movement of the volume elements through the mold and through different cooling zones - temporally variable boundary conditions are needed.

The initial condition for a newly created volume element is $$T\left({\overrightarrow{x}}_{\mathit{lag}}\left(t_{\mathit{begin}}\right)\right)=T_{\mathit{tund}}$$

wobei
λ(T)  Temperaturabhängige Wärmeleitfähigkeit in $\left[\frac{W}{\mathit{mK}}\right]$

${\frac{\partial T}{\partial n}|}_{\mathit{Oberlfläche}}$

  Temperaturgradient normal zur Oberfläche
q(t)  Spezifischer Wärmestrom zur Zeit tThe boundary condition is general $$\lambda \left(T\right)\frac{\partial T}{\partial n}{|}_{\mathit{surface}}=q\left(t\right)$$

in which
λ (T) Temperature-dependent thermal conductivity in $\left[\frac{W}{\mathit{mK}}\right]$

${\frac{\partial T}{\partial n}|}_{\mathit{Oberlfläche}}$

Temperature gradient normal to the surface
q (t) Specific heat flow at time t

In order to model the heat flow q (t), the following batch is used within the cooled mold 1 $$q\left(t\right)={\alpha }_{\mathit{mold}}\left(T_{\mathit{surf}}\left(t\right)\right)\cdot \left(T_{\mathit{surf}}\left(t\right)-T_{\mathit{mold}}\right)$$

wobei

α_mold (T_surf (t))

Wärmeabfuhrfunktion der Kokille

α _water (sw(t))

Wärmeabfuhrfunktion der Strangkühlung

sw(t)

Kühlwassermenge der Strangkühlung

a_roll (t)

Wärmeabfuhrfunktion der Stützrollen

σ

Stefan-Boltzmann Konstante

ε

der Emissionsgrad

T_surf (t)

Oberflächentemperatur des Strangs 6

T_amb

Umgebungstemperatur

Outside the mold is $$q\left(t\right)={\alpha }_{\mathit{water}}\left(\mathrm{sw}\left(t\right)\right)\cdot \left(T_{\mathit{surf}}\left(t\right)-T_{\mathit{water}}\right)+a_{\mathit{rolling}}\left(t\right)\cdot \left(T_{\mathit{surf}}\left(t\right)-T_{\mathit{rolling}}\right)+\underset{\mathit{radiation}}{\underset{\}}{\sigma \cdot \epsilon \cdot \left(T_{\mathit{surf}}{\left(t\right)}^4-{T_{\mathit{amb}}}^4\right)}}$$

in which

α _mold ( T _surf ( t ))

Heat dissipation function of the mold

α _water ( sw ( t ))

Heat dissipation function of strand cooling

sw ( t )

Cooling water quantity of the strand cooling

a _roll ( t )

Heat dissipation function of the support rollers

σ

Stefan-Boltzmann constant

ε

the emissivity

T _surf ( t )

Surface temperature of the strand 6

T _amb

ambient temperature

Since the three-dimensional heat equation is still unresolved to date, the high accuracy of the formulation of the heat _equation according to the invention with a transformed enthalpy E _trans should be checked using a one-dimensional example. The exact solution of the heat conduction equation taking into account the temperature - dependent density change (solid line) is known in the one - dimensional case and is in Fig. 3 with a mass correct formulation (dashed line) and a mass and enthalpierichtage formulation (dotted, solid line) compared. In Fig. 3 On the ordinate, the distance in strand extraction direction from the mold and on the abscissa the thickness of a metal strand in the beach edge direction is plotted. As in Fig. 3 can be seen, in a mass and enthalpierichtichtung the heat conduction equation, the actual Durcherstarrungspunkt is slightly further away from the mold than the calculated Durcherstarrungspunkt, ie. the solidification point is slightly overestimated. In comparison, the solidification point is significantly underestimated in a mass correct formulation of the heat equation, which can lead to critical situations in the continuous casting process.

LIST OF REFERENCE NUMBERS

1

mold

2

stole

3

Zumischgefäß

4

Liquid core of the strand

5

strand shell

6

strand

7

Strand support means

8th

support rollers

9

division of roles

10

cooling nozzles

11

Supply line coolant

12

Valve

13

Flow meter

14

actuator

15

process computer

16

control element

17

input unit

18

output unit

19

Discrete volume element

20

Quadrant of the strand

Claims (5)

1. Method for continuous casting of a metal strand (6) in a continuous casting plant, wherein the metal strand (6) is extracted from a cooled continuous casting mould (1) with a fluid core (4) enclosed by a strand shell (5), supported in a strand supporting device (7) arranged downstream of the continuous casting mould (1) and cooled by a coolant, wherein thermodynamic status changes of the entire metal strand (6) are calculated in a mathematical simulation model, taking into account the physical parameters of the metal, the thickness of the metal strand (6) and the continuously measured extraction speed, characterised in that in the mathematical simulation model a three-dimensional thermal conduction equation taking into account temperature-dependent changes in density of the metal strand (6) is solved numerically in real time and the cooling of the metal strand (6) is adjusted while taking into account the calculated status changes.
2. Method according to claim 1, characterised in that the metal strand (6) is embodied as a steel strand.
3. Method according to one of the preceding claims, characterised in that, in the numerical solution of the thermal conduction equation taking into account temperature-dependent changes in density of the metal strand (6), approximated equations are used for the enthalpy, which have the exact mass and the exact enthalpy for the entire metal strand (6).
4. Method according to one of the preceding claims, characterised in that the thermal conduction equation is solved numerically using a finite volume method or a finite element method.
5. Method according to one of the preceding claims, characterised in that the thermodynamic status changes are only calculated for a quarter (20) of the strand cross section as a result of the spatial symmetry.

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