EP4119247A1 - Incorporation of state-dependent density when solving a heat conduction equation - Google Patents

Abstract

At a treatment time, a treatment device (2) is intended to act on a flat, warm metal rolling stock (1) at least essentially in the thickness direction. At least for a period before the treatment time, the temporal development of a thermal state (Z) of the rolled stock (1) is modeled using a model (16) of the rolled stock (1) by iteratively solving at least one heat conduction equation. The treatment device (2) is controlled depending on the thermal state (Z) that is determined for the rolling stock (1) for the treatment time using the model (16). The density (p) of the rolling stock (1) is included in the heat conduction equation, which in turn depends on the respective thermal state (Z) of the rolling stock (1). The dependence of the density (p) on the thermal state (Z) is taken into account in the heat conduction equation by a factor (a, a') of the form a=xL⋅xBxD⋅ρ0oraʹ=1xD2⋅ρ, where p and ρ<sub>0< /sub> are densities (p, ρ₀) related to a current and a predetermined thermal state (Z, Z0) of the rolling stock (1) and the coefficients xL, xB and xD are the quotient of an extent (L, B, D, L₀, B₀, D₀) of the rolled stock (1) in the longitudinal direction, in the width direction and in the thickness direction in the respective and the predetermined thermal state (Z, Z0) of the rolling stock (1).

Description

field of technology

• wobei zu einer Behandlungszeit mittels einer Behandlungseinrichtung zumindest im wesentlichen in Dickenrichtung auf das Walzgut eingewirkt wird,
• wobei zumindest für einen Zeitraum vor der Behandlungszeit mittels eines Modells des Walzguts durch iteratives Lösen zumindest einer Wärmeleitungsgleichung die zeitliche Entwicklung eines thermischen Zustands des Walzguts modelliert wird,
• wobei eine Ansteuerung der Behandlungseinrichtung, aufgrund derer die Behandlungseinrichtung auf das Walzgut einwirkt, in Abhängigkeit von demjenigen thermischen Zustand erfolgt, der mittels des Modells für das Walzgut für die Behandlungszeit ermittelt wird,
• wobei die Dichte des Walzguts in die Wärmeleitungsgleichung eingeht,
• wobei die Dichte vom jeweiligen thermischen Zustand des Walzguts abhängt.

The present invention is based on a treatment method for a metal rolling stock, wherein the rolling stock is a flat, hot rolling stock that extends in a longitudinal direction, in a width direction and in a thickness direction,

• wherein at a treatment time the rolling stock is acted on at least essentially in the thickness direction by means of a treatment device,
• the development over time of a thermal state of the rolling stock being modeled at least for a period before the treatment time by means of a model of the rolling stock by iteratively solving at least one heat conduction equation,
• wherein the treatment device is controlled, on the basis of which the treatment device acts on the rolling stock, depending on the thermal state that is determined using the model for the rolling stock for the treatment time,
• where the density of the rolling stock is included in the heat conduction equation,
• where the density depends on the respective thermal state of the rolling stock.

The present invention is also based on a control program for a control device of a treatment device for treating a metal rolling stock, the control program comprising machine code which can be processed by the control device, the processing of the machine code by the control device causing the control device operates the treatment facility according to such a treatment method.

The present invention is also based on a control device of a treatment device for treating a rolling stock made of metal, the control device being programmed with such a control program so that the control device operates the treatment device according to such a treatment method.

• wobei die Behandlungsanlage eine Behandlungseinrichtung aufweist, mittels derer zumindest im wesentlichen in Dickenrichtung auf das Walzgut einwirkbar ist,
• wobei die Behandlungsanlage eine Steuereinrichtung aufweist, von der zumindest die Behandlungseinrichtung gesteuert wird,
• wobei die Steuereinrichtung als entsprechende Steuereinrichtung ausgebildet ist, so dass die Steuereinrichtung die Behandlungseinrichtung gemäß einem derartigen Behandlungsverfahren betreibt.

The present invention is also based on a treatment system for treating a rolled stock made of metal, the rolled stock being a flat, warm rolled stock which extends in a longitudinal direction, in a width direction and in a thickness direction,

• wherein the treatment plant has a treatment device, by means of which the rolling stock can be acted on at least essentially in the thickness direction,
• wherein the treatment system has a control device, by which at least the treatment device is controlled,
• wherein the control device is designed as a corresponding control device, so that the control device operates the treatment device according to such a treatment method.

State of the art

In the production of flat rolled stock, ie in connection with casting, rough rolling, finish rolling and cooling, it is often necessary to know exactly the temperature or, in general, the thermal state of the flat rolled stock at specific times. However, it is often not possible to measure the temperature and thus the thermal state. For this reason, the thermal state of the flat rolling stock is modeled accordingly.

A heat conduction equation is often used to model properly. The heat conduction equation is a differential equation that has to be solved iteratively in small time steps. Depending on the procedure, the heat conduction equation can be applied in different ways. Depending on the situation, it may also be necessary to iteratively solve a phase transformation equation in parallel to solving the heat conduction equation.

Heat conduction equations are known in various configurations. For example, it is known to apply the heat conduction equation one-dimensionally or three-dimensionally. In the case of a one-dimensional approach, the heat conduction equation is solved only in the direction of the thickness of the flat rolling stock. Longitudinal and latitudinal heat flow is neglected. In the following - in the sense of an incomplete list - some possible one-dimensional approaches for the heat conduction equation are listed and explained. The corresponding three-dimensional approaches are not explained separately, but are (at least mostly) also specified in the prior art.

anzusetzen. Hierbei sind T die Temperatur, λ die Wärmeleitfähigkeit, ρ die Dichte und c_P die Wärmekapazität des Walzguts. t und s sind die Zeit und der Ort in Dickenrichtung des Walzguts. Dieser Ansatz ist beispielsweise in dem Fachbuch "Einführung in partielle Differenzialgleichungen" von Aslak Tveito und Ragnar Winther, Springer-Verlag 2002, erläutert. Dieser Ansatz ist linear, basiert auf der Temperatur und arbeitet ohne Wärmequellen.For example, it is possible to write the heat conduction equation in the form $$\frac{\partial T}{\partial t}=\frac{\lambda }{\rho \cdot c_P}\cdot \frac{{\partial }^{2}T}{\partial s^2}$$

to set. Here T is the temperature, λ is the thermal conductivity, ρ is the density and c _P is the heat capacity of the rolling stock. t and s are the time and place in the thickness direction of the rolled stock. This approach is explained, for example, in the specialist book "Introduction to Partial Differential Equations" by Aslak Tveito and Ragnar Winther, Springer-Verlag 2002. This approach is linear, based on temperature and works without heat sources.

Alternatively, if the heat capacity is used as a function of temperature, Equation 1 is rewritten as follows: $$\rho \cdot c_P\left(T\right)\cdot \frac{\partial T}{\partial t}=\frac{\partial }{\partial s}\left(\lambda \left(T\right)\cdot \frac{\partial T}{\partial s}\right)$$

The difference to Equation 1 is that the heat capacity c _P is now temperature-dependent and thus indirectly variable over time, and a non-linear behavior of the rolling stock can be modeled. The thermal conductivity can be state dependent. This approach is also explained in the referenced book by Aslak Tveito and Ragnar Winther. It is non-linear and also based on temperature.

anzusetzen. Hierbei sind - zusätzlich zu den bereits erläuterten Größen - H die Enthalpie und p ein Phasenzustand. Der Phasenzustand kann skalar oder vektoriell sein. Mit Q werden Wärmequellen oder Wärmesenken modelliert. Dieser Ansatz wird beispielsweise in der EP 1 397 523 A1 , der EP 1 576 429 A1 und der EP 1 711 868 B1 erläutert. Dieser Ansatz ist nichtlinear, arbeitet mit der Enthalpie und der Phasenumwandlung sowie mit Quellen. Die Temperatur ist eine aus der Enthalpie und dem Phasenzustand abgeleitete Größe. Bei diesem Ansatz können Wärmequellen und Wärmesenken mit berücksichtigt werden.Alternatively, it is possible, for example, to use the heat conduction equation in the form $$\rho \cdot \frac{\partial H}{\partial t}=\frac{\partial }{\partial s}\left(\lambda \left(H,p\right)\cdot \frac{\partial T\left(H,p\right)}{\partial s}\right)+Q\left(s\right)$$

to set. Here - in addition to the quantities already explained - H is the enthalpy and p is a phase state. The phase state can be scalar or vectorial. Heat sources or heat sinks are modeled with Q. This approach is used, for example, in EP 1 397 523 A1 , the EP 1 576 429 A1 and the EP 1 711 868 B1 explained. This approach is non-linear, works with the enthalpy and the phase transition and with sources. The temperature is a variable derived from the enthalpy and the phase state. With this approach, heat sources and heat sinks can be taken into account.

The latter approach can be extended to the effect that the concentration of a dissolved alloying element in a phase (particularly in the case of steel, the concentration of carbon in the austenite phase) is also taken into account will. This procedure is in the EP 1 910 951 B1 explained in detail.

anzusetzen. Die zugehörigen Größen sind bereits erläutert. Dieser Ansatz ist beispielsweise in der WO 2017/092 967 A1 erwähnt. In der WO 2017/092 967 A1 ist weiterhin ausgeführt, dass für die einzelnen Phasen jeweils eine Dichte ermittelt werden kann, dass Phasengrenzen zwischen den Phasen ermittelt werden können und auf Basis der ermittelten Dichten und der ermittelten Phasengrenzen eine Dichteverteilung ermittelt werden kann. Die genaue Kenntnis der Dichteverteilung soll eine genauere Ermittlung der Temperaturverteilung ermöglichen.As a last example, the approach should be mentioned, the heat conduction equation in the form $$\rho \cdot c_P\cdot \frac{\partial T}{\partial t}-\frac{\partial }{\partial s}\left(\lambda \cdot \frac{\partial T}{\partial s}\right)=Q$$

to set. The associated variables have already been explained. This approach is for example in the WO 2017/092 967 A1 mentioned. In the WO 2017/092 967 A1 it is further stated that a density can be determined for each of the individual phases, that phase boundaries between the phases can be determined and a density distribution can be determined on the basis of the densities determined and the phase boundaries determined. Exact knowledge of the density distribution should enable a more precise determination of the temperature distribution.

ermittelt. Hierbei ist ρ - wie zuvor - die Dichte, f_i sind Anteile von Phasen und ρ_i sind die Dichten der Phasen.In the technical paper " Calculation of Thermophysical Properties of Carbon and Low Alloyed Steels for Modeling of Solidification Processes by Jyrki Miettinen and Seppo Louhenkilpi, Metallurgical and Materials Transactions, Volume 25B, December 1994, pages 909 to 916 , various heat conduction equations are also mentioned in connection with solidification processes. The density is formulated as a state-dependent variable. Specifically, it becomes according to the relationship $$\rho =\frac{1}{\mathrm{\Sigma }\frac{f_i}{{\rho }_i}}$$

determined. Here ρ is - as before - the density, f _i are fractions of phases and ρ _i are the densities of the phases.

Summary of the Invention

Regardless of the specific heat conduction equation used, the heat conduction equation (among other things) always includes the density. This applies regardless of whether one of the approaches explained above is used for the heat conduction equation or another approach. It also applies regardless of whether the heat conduction equation is set as one-dimensional or multi-dimensional, and regardless of whether the heat conduction equation is set as linear or non-linear.

In the vast majority of approaches, it is assumed that the density of the material does not change, so that when solving the heat conduction equation, the density can be used as a constant. the WO 2017/092 967 A1 mentions that density can be used as a variable. However, there are no explanations as to how this variable density should be taken into account in the heat conduction equation. The approach of Miettinen and Louhenkilpi only applies specifically to solidification processes and is not applicable to rolling stock where the metal has already solidified. The reason for this is that a change in the density of a solid always entails a change in its dimensions, which must be taken into account when discretizing the heat conduction equation, just like the change in density itself. In the technical article mentioned, on the other hand, a change in the direction of thickness caused by rollers that are in contact with the rolling stock is prevented in particular.

The object of the present invention is to create possibilities by means of which the state-dependent density of the rolling stock can be correctly taken into account and the accuracy when solving the heat conduction equation can thus be improved.

The object is achieved by an operating method with the features of claim 1. Advantageous configurations of the operating method are the subject matter of dependent claims 2 to 5.

berücksichtigt wird, wobei ρ und ρ₀ auf einen aktuellen und einen vorbestimmten thermischen Zustand des Walzguts bezogene Dichten des Walzguts sind und die Koeffizienten xL, xB und xD sich als Quotient einer Erstreckung des Walzguts in Längsrichtung, in Breitenrichtung und in Dickenrichtung bei dem jeweiligen und dem vorbestimmten thermischen Zustand des Walzguts ergeben.According to the invention, a treatment method of the type mentioned is designed in that the dependence of the density of the rolling stock on the respective thermal state of the rolling stock in the heat conduction equation by a factor of the form $$a=\frac{\mathit{xL}\cdot \mathit{xB}}{\mathit{xD}\cdot {\rho }_0}\mathrm{or}\mathit{aʹ}=\frac{1}{{\mathit{<figure-callout\; id="2"\; label="xD"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">xD</figure-callout>}}^2\cdot \rho }$$

is taken into account, where ρ and ρ ₀ are densities of the rolling stock related to a current and a predetermined thermal state of the rolling stock and the coefficients xL, xB and xD are the quotient of an extension of the rolling stock in the longitudinal direction, in the width direction and in the thickness direction for the respective and result from the predetermined thermal state of the rolling stock.

The dependence of the heat conduction equation on the density of the rolling stock can be fully taken into account by the factor a or a'. Beyond the factor a or a', the heat conduction equation is therefore independent of the density of the rolling stock.

The factors a and a' will be derived later for a very specific heat conduction equation. However, they are completely independent of the specific heat conduction equation used. So any of Equations 1 through 4 can be used, even in their multidimensional form. Likewise, other heat conduction equations can also be used. If required, it is also possible to solve the heat conduction equation used with simultaneous coupling with a phase transformation equation and thereby to take into account the heat of transformation occurring during the phase transformation.

In the general case, the coefficients xL, xB and xD must be known individually and individually. However, the density is often isotropic. In this case changes in density are also isotropic. Thus, only the density needs to be known as such being. Because in this case it is possible to determine the coefficients xL, xB and xD in such a way that they are equal to each other and their product is equal to the quotient of the density, which is dependent on the respective thermal state of the rolling stock, and the density related to the predetermined thermal state of the rolling stock normalized density. Due to the equality of the coefficients xL, xB and xD, the factor a can increase $$a=\frac{\mathit{xD}}{{\rho }_0}$$

be simplified. Instead of the coefficient xD, the coefficient xL or the coefficient xB or the cube root of the density change could of course also be used (due to the equality).

If the factor a' is used, it is only necessary to determine the coefficient xD. In this case, the coefficient xD can be determined such that it is equal to the third root of the quotient of the density dependent on the respective thermal state of the rolling stock and the normalized density related to the predetermined thermal state of the rolling stock.

In some cases it can be sufficient to model the temporal development of the thermal state of the rolling stock offline. As a rule, however, the temporal development of the thermal state of the rolling stock is modeled online - for example as part of a setup calculation - or even in real time.

The type of treatment can be as needed. For example, it is possible for the treatment device to roll the rolling stock so that the thickness of the rolling stock is smaller after the treatment device has acted on the rolling stock than before the treatment device has acted on the rolling stock. The thermal conductivity determined by the model In this case, the state of the rolling stock can be used, for example, within the scope of determining the deformation resistance of the rolling stock and thus determining the required rolling force.

Often, the treatment device is used to influence the rolling stock purely thermally, without forming the rolling stock. Such a thermal influencing of the rolling stock can, if required, be, for example, heating (for example inductive heating) before a roughing train or before a finishing train. It can also be an effect in which cooling occurs as an unavoidable side effect, for example during descaling of the rolling stock. Above all, however, it can be intentional cooling. In this context, for example, interstand cooling (ie cooling between individual rolling processes in a multi-stand rolling train) or cooling in a cooling section downstream of a rolling device can be mentioned.

The object is also achieved by a control program with the features of claim 6. According to the invention, the execution of the control program causes the control device to operate the treatment device according to a treatment method according to the invention.

The object is also achieved by a control device with the features of claim 7. According to the invention, the control device is programmed with a control program according to the invention, so that the control device operates the treatment device according to a treatment method according to the invention.

The object is also achieved by a treatment plant with the features of claim 8. According to the invention, the control device is designed as a control device according to the invention, so that the control device is the treatment device operates according to a treatment method according to the invention.

Brief description of the drawings

FIG 1

eine Behandlungsanlage von der Seite,

FIG 2

ein flaches Walzgut von oben,

FIG 3

eine Warmwalzanlage einschließlich Kühlstrecke,

FIG 4

ein Ablaufdiagramm,

FIG 5

ein Ablaufdiagramm,

FIG 6

eine Wärmeleitungsgleichung,

FIG 7

eine Steuereinrichtung,

FIG 8

ein Volumenelement,

FIG 9

eine Wärmeleitungsgleichung,

FIG 10

eine Modifikation der Steuereinrichtung,

FIG 11

eine weitere Modifikation der Steuereinrichtung,

FIG 12

ein Ablaufdiagramm,

FIG 13

eine Wärmeleitungsgleichung,

FIG 14

eine Modifikation der Steuereinrichtung und

FIG 15

eine weitere Modifikation der Steuereinrichtung.

The characteristics, features and advantages of this invention described above, and the manner in which they are achieved, will become clearer and more clearly understood in connection with the following description of the exemplary embodiments, which are explained in more detail in connection with the drawings. This shows in a schematic representation:

FIG 1

a treatment plant from the side,

FIG 2

a flat rolling stock from above,

3

a hot rolling mill including a cooling section,

FIG 4

a flowchart,

5

a flowchart,

6

a heat conduction equation,

FIG 7

a control device,

8

a volume element,

9

a heat conduction equation,

10

a modification of the control device,

11

another modification of the control device,

12

a flowchart,

13

a heat conduction equation,

14

a modification of the control device and

15

another modification of the control device.

Description of the embodiments

According to FIG 1 a treatment plant for rolling stock 1 has a treatment device 2 . The rolling stock 1 can be acted on by means of the treatment device 2 .

The rolling stock 1 consists of metal. The rolling stock 1 usually consists of steel. However, it can also consist of another metal, for example aluminum or copper. The rolling stock 1 extends - see FIG 2 - in a longitudinal direction over a overall length L and in a width direction over an overall width B. Furthermore, the rolling stock 1 also extends in a thickness direction—see FIG FIG 1 - Over a thickness D. The thickness D is smaller than the width B, usually considerably smaller. The width B is usually smaller than the length L. Typical values for the thickness D are in the range from less than 1 mm to 250 mm, sometimes a little more. Typical values for the width B are in the range between 500 mm and 2500 mm, in some cases even slightly more. The length L is several meters, for example several 100 m or even up to more than 1000 m. The rolling stock 1 is therefore a flat rolling stock 1. Furthermore, the rolling stock 1 is a warm rolling stock 1.

The treatment device 2 can act on the rolling stock 1 at least essentially in the thickness direction. This is discussed below in connection with 3 explained in more detail.

According to 3 For example, the rolling stock 1 can first be rough-rolled in a roughing mill, then finish-rolled in a finishing train, then cooled in a cooling section downstream of the finishing train and finally coiled into a bundle. The roughing mill has at least one roughing stand 3 for roughing the rolling stock 1 . In an analogous manner, the finishing train for finish-rolling the rolling stock 1 has at least one finish-rolling stand 4 . The cooling section has at least one application device 5, by means of which the rolling stock 1 can be subjected to cooling water.

A heating device 6 (in particular an induction heater) and/or a descaling device 7 can be arranged upstream of the roughing mill. In an analogous manner, a heating device 8 (in particular an induction heater) and/or a descaling device 9 can be arranged upstream of the finishing train. In the case of a multi-stand finishing train, 4 intermediate stand cooling systems 10 can also be arranged between the finishing rolling stands, by means of which the rolling stock 1 is between the individual finishing rolling stands 4 can be charged with cooling water.

Each of the components 3 to 10 mentioned can be a treatment device 2 within the meaning of the present invention. In the case of the roll stands 3, 4, the rolling stock 1 is rolled. In this case, the thickness D of the rolling stock 1 after the action of the treatment device 2, 3, 4 on the rolling stock 1 is generally smaller than before the action of the treatment device 2, 3, 4 on the rolling stock 1. Equality only exists in exceptional cases when the rolling stock 1 passes through the treatment device 2, 3, 4 without forming. In the case of the application device 5, the heating devices 6, 8, the descaling devices 7, 9 and the interstand cooling 10, the action of the treatment device 2, 5 to 10 on the rolling stock 1 has a purely thermal effect on the rolling stock 1 without forming. In the case of the heating devices 6, 8, the purely thermal influence is a heating of the rolling stock 1. In the case of the application devices 5, the descaling devices 7, 9 and the interstand cooling in 10, the purely thermal influence is a cooling of the rolling stock 1.

In the following, only the reference number 2 is always used for the treatment device. The full listing of reference numerals 2, 3, 4, etc. would only bloat and obscure the text without contributing to an understanding of the present invention.

The treatment device 2 (and optionally also other components of the treatment system) are in accordance with FIG 1 controlled by a control device 11. The control device 11 is programmed with a control program 12 . The control program 12 includes machine code 13 which can be processed by the control device 11 . The programming of the control device 11 with the control program 12 (or equivalently the processing of the machine code 13 by the control device 11) causes the control device 11 to Treatment device 2 operates according to a treatment method, as below - initially in connection with FIG 4 - is explained in more detail.

According to FIG 4 an initial state ZA of the rolling stock 1 becomes known to the control device 11 in a step S1. The initial state ZA can, under certain circumstances, relate to the entire rolling stock 1 . Alternatively, he himself - see FIG 2 - Refer to a single section 14 of the rolling stock 1. It is assumed below that the initial state ZA relates to an individual section 14 of the rolling stock 1 . This also represents the general case, since in the case of a uniform consideration of the entire rolling stock 1, only the number of sections 14 has to be reduced accordingly, namely to a single section 14.

A thermal state of the section 14 is determined by the initial state ZA. In particular, at least the initial temperature T of the corresponding section 14 is determined by it—directly or indirectly. If necessary, a phase state p can also be determined. For example, the initial state ZA can contain the initial temperature or the initial enthalpy of the section 14, in both cases with or without phase components or at least one phase component.

For example, the rolling stock 1 as shown in the 1 to 3 be conveyed at a constant or variable speed v, the speed v running in the longitudinal direction of the rolling stock 1. In this case, for example, the initial state ZA can be recorded and fed to the control device 11 with a fixed time cycle, for example by means of a corresponding measuring device 15 (for example by means of a temperature measuring station). In this case, the control device 11 assigns the initial state ZA to the corresponding section 14 . However, other procedures are also possible. In the case of a fixed time cycle is the length of the respective section 14, to which the respective initial state ZA refers, is determined by the speed v during the respective timing cycle and the timing cycle itself. The timing is typically between 0.1 s and 0.5 s, in particular between 0.2 s and 0.4 s, for example 0.3 s.

In a step S2, the control device 11 sets a current state Z of the section 14 equal to the initial state ZA. The current state Z is therefore also a thermal state.

In a step S3, the control device 11 updates the current state Z of the section 14. In particular, in step S3 the control device 11 sets in a model 16 of the rolling stock 1 (see FIG 1 ) provides at least one heat conduction equation for section 14 and solves the heat conduction equation for a single time step. This will be explained later in detail. If required, the control device 11 can also apply and solve a phase transformation equation for the section 14 in step S3.

If necessary, the control device 11 implements a path tracking for the section 14 in a step S4. A path tracking and its implementation is generally known to those skilled in the art.

In a step S5, the control device 11 checks whether a treatment time has been reached at which the section 14 of the rolling stock 1 should be treated in the treatment device 2, ie the treatment device 2 should act on the section 14 in the thickness direction. For clarification: The treatment time is not a period of time detached from an absolute time, but a fixed point in time or a fixed period of time. The term "treatment time" therefore does not mean that the section 14 should be acted on in the treatment device 2 for--for example--5 s, regardless of when this happens. The term "treatment time" rather means that in the treatment facility 2 at a specific point in time - for example exactly at 1:39:22 p.m. - the section 14 should be acted on or from the specific point in time for a predetermined period of time - for example for 5 s - on the section 14 is to be acted upon.

If the treatment time has not yet been reached, the control device 11 goes back to step S3. If, on the other hand, the treatment time has been reached, the control device goes to a step S6.

In step S6, the control device 11 determines a control A for the treatment device 2 depending on the current state Z, which was determined using the model 16 for the section 14 for the treatment time. In the case of a rolling process, when determining the control A, for example, the Material strength of section 14 are also taken into account, as they result from the current state Z (among other things). In the case of a purely thermal influence, the extent of the influence—for example the amount of coolant that is to be applied to section 14—can be determined as a function of the current state Z.

In a subsequent step S7, the control device 11 controls the treatment device 2 according to the control A determined. Due to the control A, the treatment device 2 acts on the rolling stock 1 in the thickness direction.

As a result, the procedure of FIG 4 the development over time of the current state Z of the rolling stock 1 is thus modeled at least for a period before the treatment time by means of the model 16 by iteratively solving at least one heat conduction equation. If necessary, as already mentioned, iteratively and with mutual coupling a phase transformation equation can be solved simultaneously with the heat conduction equation.

In many cases, the procedure of FIG 4 accordingly 5 added. In this case, step S7, in which the action is taken on section 14, is followed by steps S11 to S16.

In step S11, the control device 11 updates the current state Z of the section 14 according to the control A.

The control device 11 updates the current state Z in a step S12. The content of step S12 corresponds to step S3. Furthermore, if necessary, the control device 11 implements path tracking for the corresponding section 14 in a step S13.

In a step S14, the control device 11 checks whether the section 14 has reached a detection point at which an actual thermal state ZT of the section 14 is detected by means of a further measuring device 17. The measuring device 17 can be a temperature measuring station, for example.

If the detection point has not yet been reached, the control device 11 goes back to step S12. If, on the other hand, the detection point is reached, the control device 11 goes to a step S15. In step S15 the control device 11 accepts the actual thermal state ZT of the section 14 . In a step S16, the control device 11 then compares the last determined current state Z with the actual thermal state ZT recorded by measurement. Based on the comparison, the control device 11 takes further measures in step S16. For example, it can adapt the model 16 or track the control A in the sense of a target/actual regulation.

angesetzt werden. In Gleichung 8 sind H die Enthalpie (oder Energiedichte), t die Zeit, s die Ortsvariable in Dickenrichtung, λ die Wärmeleitfähigkeit, ρ die Dichte und T die Temperatur. Die Argumente der Variablen sind in Gleichung 8 nicht mit angegeben, da es im Rahmen der vorliegenden Erfindung auf sie nicht ankommt. In Verbindung mit dem Ansatz gemäß Gleichung 8 wird die vorliegende Erfindung nachstehend erläutert. Die erfindungsgemäße Vorgehensweise ist aber unabhängig von dem getroffenen Ansatz auch für andere Wärmeleitungsgleichungen gültig.In the procedures of the prior art, the heat conduction equation that is solved in steps S3 and S12, for example as shown in FIG 6 in the model 16 an equation of the form $$\frac{\mathit{i.e}}{\mathit{German}}=\frac{\mathrm{i.e}}{\mathit{ds}}\left(\frac{\lambda }{\rho }\cdot \frac{\mathit{dT}}{\mathit{ds}}\right)$$

be scheduled. In Equation 8, H is the enthalpy (or energy density), t is the time, s is the thickness variable, λ is the thermal conductivity, ρ is the density, and T is the temperature. The arguments of the variables are not given in Equation 8, since they are not important in the context of the present invention. In connection with the approach according to equation 8, the present invention is explained below. However, the procedure according to the invention is also valid for other heat conduction equations, regardless of the approach taken.

As can be seen, the density ρ and the thermal conductivity λ of the rolling stock 1 go into the heat conduction equation. The thermal conductivity λ is generally dependent on the respective thermal state Z of the rolling stock 1 (or the corresponding section 14). The control device 11 is therefore in accordance with FIG 7 the thermal conductivity λ for a large number of possible thermal states Z is supplied. The density ρ is assumed to be constant and is therefore supplied to the control device 11 as a constant. This procedure is known in the prior art and is not (yet) the subject of the present invention. If the accuracy requirements are not too high, this procedure is completely satisfactory.

As is well known, the density ρ is not constant, but varies at least as a function of the temperature, and often also as a function of the phase state p. The density ρ, like the thermal conductivity λ, is therefore dependent on the respective thermal state Z of the rolling stock 1. The fact that the variability of the density ρ in equation 8 is not taken into account leads to inaccuracies in the modelling. The - at least extensive - compensation of these inaccuracies is the subject of the present invention.

definiert ist.If the rolling stock 1 has a predetermined thermal state Z0, the rolling stock 1 has a specific density ρ ₀ . This density ρ ₀ is referred to below as normalized density. If the current state Z of the rolling stock 1 deviates from the predetermined thermal state Z0, then the current density ρ of the rolling stock 1 usually also differs from the normalized density ρ ₀ of the rolling stock 1. The deviation can be described by a value x, where x increases $$x=\frac{{\rho }_0}{\rho }$$

is defined.

$$B=\mathit{xB}\cdot B_0$$

$$D=\mathit{xD}\cdot D_0$$

wobei L, B und D die Länge, die Breite und die Dicke des Walzguts 1 bei dem jeweiligen Zustand Z und L₀, B₀ und D₀ die Länge, die Breite und die Dicke des Walzguts 1 bei dem vorbestimmten thermischen Zustand Z0 sind. Die auf den vorbestimmten thermischen Zustand Z0 bezogenen Größen werden nachfolgend als normierte Länge, normierte Breite und normierte Dicke bezeichnet.The change in density ρ corresponds to a change in volume. The relationships thus result for the length L, the width B and the thickness D of the rolling stock 1 $$L=\mathit{xL}\cdot L_0$$

$$B=\mathit{xB}\cdot B_0$$

$$D=\mathit{xD}\cdot D_0$$

where L, B and D are the length, width and thickness of the rolled stock 1 at the respective state Z and L ₀ , B ₀ and D ₀ are the length, width and thickness of the rolled stock 1 at the predetermined thermal state Z0. The quantities related to the predetermined thermal state Z0 are referred to below as normalized length, normalized width and normalized thickness.

gelten.Since the mass of the rolling stock 1 does not change and the density ρ is defined as the quotient of mass and volume, the relationship must continue to exist $$x=\mathit{xL}\cdot \mathit{xB}\cdot \mathit{xD}$$

be valid.

The following problem is associated with the changes, in particular of the thickness D, but also of the length L and the width B: In practice, the heat conduction equation is solved for predetermined interpolation points. The interpolation points are specified once and have specific (small) distances ds ₀ from one another when specified, particularly in the direction of thickness. With the change in density ρ, however, the position of the interpolation points also changes. This also changes the distances ds between the interpolation points. With each new determination of the state Z, the distances ds would therefore have to be updated. This proves to be unwieldy in practice. The modeling is considerably simplified if the distances ds ₀ can be retained uniformly, i.e. the distance ds ₀ is used throughout the calculation. This affects the heat conduction equation.

In order to explain the effects on the heat conduction equation, a small volume element 18 is considered below, ie a volume element 18 which is defined according to FIG 8 (see there on the left) has the normalized length L ₀ , the normalized width B ₀ and the normalized thickness ds ₀ . In the currently considered state Z, however, the volume element 18 (see in 8 right) actually has the length L, the width B and the thickness ds.

So that the interpolation points at which the heat conduction equation is solved can be retained unchanged, regardless of the state Z considered in each case, a (geometric) transformation of the volume element 18 considered from the normalized dimensions L ₀ , B ₀ and ds ₀ to the dimensions L, B , ds made. In order to be able to carry out this transformation, the heat conduction equation must be suitably adjusted. To determine this adjustment, see below various quantities can be mentioned which refer to the normalized dimensions L ₀ , B ₀ and ds ₀ of the volume element 18 with the index "0" and to the actual dimensions L, B, ds of the volume element 18 without the index "0". For the sake of brevity, instead of the phrases "related to the normalized dimensions L ₀ , B ₀ , ds ₀ " and "related to the dimensions L, B, ds", the phrases "in the untransformed state" and "in the transformed state" are used only briefly below. used.

The energy density H (unit: J/kg) does not initially change as a result of the geometric transformation. So it applies $$H=H_0$$

The temperature T does not change as a result of the geometric transformation. It is therefore valid $$T=T_0$$

However, due to the geometric transformation, the thickness ds changes according to the coefficient xD. This also changes the temperature gradient. So it applies $$\frac{\mathit{dT}}{\mathit{ds}}=\frac{1}{\mathit{xD}}\cdot \frac{\mathit{dT}}{{\mathit{ds}}_0}$$

Furthermore, the change in density ρ affects the thermal conductivity λ. For physical reasons - at least for small changes in the density ρ - there must be a proportionality. So it applies $$\lambda =\frac{{\lambda }_0}{x}$$

The heat flow density j is the product of the thermal conductivity λ and the temperature gradient. So it applies $$j=\lambda \cdot \frac{\mathit{dT}}{\mathit{ds}}=\frac{{\lambda }_0}{x}\cdot \frac{\mathit{dT}}{\mathit{xD}\cdot {\mathit{ds}}_0}=\frac{1}{x\cdot \mathit{xD}}\cdot {\lambda }_0\cdot \frac{\mathit{dT}}{{\mathit{ds}}_0}=\frac{1}{x\cdot \mathit{xD}}\cdot j_0$$

Inserting the transformed variables into the heat conduction equation thus results in $$\frac{{\mathit{i.e}}_0}{\mathit{German}}=\frac{\mathit{i.e}}{\mathit{German}}=\frac{\mathrm{i.e}}{\mathit{xD}\cdot {\mathit{ds}}_0}\left(\frac{x\cdot {\lambda }_0}{x\cdot {\rho }_0}\cdot \frac{\mathit{dT}}{\mathit{xD}\cdot {\mathit{ds}}_0}\right)=\frac{1}{{\mathit{<figure-callout\; id="2"\; label="xD"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">xD</figure-callout>}}^2}\frac{\mathrm{i.e}}{{\mathit{ds}}_0}\left(\frac{{\lambda }_0}{{\rho }_0}\cdot \frac{\mathit{dT}}{{\mathit{ds}}_0}\right)$$

In practice, according to the controller 11 9 (and also according to FIG 7 ) the thermal conductivity λ is specified as a function of the state Z. The control device 11 therefore does not "know" the thermal conductivity λ ₀ in the untransformed state, but rather the thermal conductivity λ in the transformed state. Based on Equation 17, we get - see also 9 - the equation $$\frac{{\mathit{i.e}}_0}{\mathit{German}}=\frac{\mathit{i.e}}{\mathit{German}}=\frac{\mathrm{i.e}}{\mathit{xD}\cdot {\mathit{ds}}_0}\left(\frac{x\cdot \lambda }{{\rho }_0}\cdot \frac{\mathit{dT}}{\mathit{xD}\cdot {\mathit{ds}}_0}\right)=\frac{\mathit{xL}\cdot \mathit{xB}}{\mathit{xD}}\cdot \frac{\mathrm{i.e}}{{\mathit{ds}}_0}\left(\frac{\lambda }{{\rho }_0}\cdot \frac{\mathit{dT}}{{\mathit{ds}}_0}\right)$$

This is a heat conduction equation derived from Equation 8, in which the dependence of the density ρ on the current state Z is correctly taken into account and which can consequently be solved in steps S3 and S12.

ergibt.In order to take into account the state-dependent density ρ, one must not simply insert the density ρ, which is dependent on the current state Z, into the otherwise unchanged heat conduction equation. Rather, one must make additional corrections. For example, one can calculate according to Equation 20 with a normalized density ρ ₀ - i.e. the density ρ ₀ related to the predetermined thermal state Z0 - and take into account the influence of the density ρ dependent on the state Z by using a factor a dependent on the current state Z, where the factor a increases $$a=\frac{\mathit{xL}\cdot \mathit{xB}}{\mathit{xD}\cdot {\rho }_0}$$

results.

In practice, it is thus possible, the control device 11 according to 10 In addition to the thermal conductivity λ (dependent on the current state Z), the normalized density ρ ₀ and the coefficients xL, xB and xD (also dependent on the current state Z) must be specified. The thermal conductivity λ and the coefficients xL, xB and xD are specified, for example, for predefined states between which interpolation is carried out. This is well known to those skilled in the art and need not be explained in detail. The normalized density ρ ₀ and the coefficients xL, xB and xD also implicitly give the density ρ that is dependent on the current state Z, even if the density ρ that is dependent on the current state Z is no longer required to solve the heat conduction equation itself.

In practice, the rolling stock 1 often behaves isotropically. This applies not only, but also to the density ρ. In this case it is possible that the control device 11 according to 11 in addition to the thermal conductivity λ (dependent on the current state Z), the density ρ (also dependent on the current state Z) is specified. In this case, the control device 11, as shown in FIG 12 (for example before the execution of steps S1 to S7) first determine the normalized density ρ ₀ independently in a step S21. For example, the control device 11 can determine, as the normalized density ρ ₀ , the greatest density p, the smallest density ρ or a value between the greatest and the smallest density ρ from the predefined densities ρ. Then, in a step S22, the control device 11 can determine the corresponding value x for the respective state Z by forming the quotient according to Equation 9. Furthermore, since the product of the coefficients xL, xB and xD must be equal to the value x according to Equation 13 and the coefficients xL, xB and xD must have the same value as one another due to the isotropy, the control device 11 finally, in a step S23, also determine the coefficient xD (and also the coefficients xL and xB).

vereinfacht werden. Der Faktor a vereinfacht sich somit zu $$a=\frac{\mathit{xD}}{{\rho }_0}$$

Due to the equality of the coefficients xL, xB and xD, the heat conduction equation solved in steps S3 and S12 can also be used $$\frac{{\mathit{i.e}}_0}{\mathit{German}}=\frac{\mathit{i.e}}{\mathit{German}}=\frac{\mathrm{i.e}}{\mathit{xD}\cdot {\mathit{ds}}_0}\left(\frac{x\cdot \lambda }{{\rho }_0}\cdot \frac{\mathit{dT}}{\mathit{xD}\cdot {\mathit{ds}}_0}\right)=\mathit{xD}\cdot \frac{\mathrm{i.e}}{{\mathit{ds}}_0}\left(\frac{\lambda }{{\rho }_0}\cdot \frac{\mathit{dT}}{{\mathit{ds}}_0}\right)$$

be simplified. The factor a thus simplifies to $$a=\frac{\mathit{xD}}{{\rho }_0}$$

If the rolling stock 1 behaves isotropically, it is still in the case of the embodiment according to FIG 10 it is not necessary to specify the coefficients xL, xB and xD individually. Since the coefficients xL, xB and xD are of equal size in the case of an isotropic behavior, it is sufficient to specify one of the coefficients xL, xB and xD individually.

Instead of the density ρ ₀ related to the predetermined thermal state Z0, one can also use the density ρ dependent on the current state Z. In this case, equation 20 transforms (see also 13 ) by inserting equation 9 $$\frac{{\mathit{i.e}}_0}{\mathit{German}}=\frac{\mathit{i.e}}{\mathit{German}}=\frac{\mathrm{i.e}}{\mathit{xD}\cdot {\mathit{ds}}_0}\left(\frac{{\lambda }_0}{{\rho }_0}\cdot \frac{\mathit{dT}}{\mathit{xD}\cdot {\mathit{ds}}_0}\right)=\frac{1}{{\mathit{<figure-callout\; id="2"\; label="xD"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">xD</figure-callout>}}^2}\cdot \frac{\mathrm{i.e}}{{\mathit{ds}}_0}\left(\frac{\lambda }{\rho }\cdot \frac{\mathit{dT}}{{\mathit{ds}}_0}\right)$$

Equation 24 is very similar to Equation 19, but not identical to Equation 19. The difference is that in Equation 19 the untransformed thermal conductivity λ ₀ and the normalized density ρ ₀ of the predetermined state are used, while in Equation 24 the thermal conductivity λ and the density ρ as given to the controller 11 are used.

ergibt.As an alternative to using the normalized density ρ ₀ , one can also calculate with the actual density ρ dependent on the state Z, if one also takes into account a factor a' dependent on the current state Z, with the factor a' increasing to $$\mathit{aʹ}=\frac{1}{{\mathit{<figure-callout\; id="2"\; label="xD"\; filenames="imgf0001.png,imgf0003.png"\; state="\{\{state\}\}">xD</figure-callout>}}^2\cdot \rho }$$

results.

In order to be able to determine the factor a′, the control device 11 according to 14 in addition to the thermal conductivity λ and the density ρ, the coefficient xD can be specified. If the rolling stock 1 behaves isotropically, it is still possible to control the control device 11 as in 11 only to specify the thermal conductivity λ and the density ρ, since the control device 11 can independently determine the coefficient xD in this case. Alternatively it is according to 15 possible to specify the coefficient xD for the control device 11 in addition to the thermal conductivity λ and the normalized density ρ ₀ .

In order to show the correct consideration of the density ρ (which is dependent on the current state Z), a very specific heat conduction equation was assumed above, namely the heat conduction equation according to Equation 8. As already mentioned, however, the way in which it is taken into account is independent of the heat conduction equation actually used . Depending on the procedure, factor a or factor a' must always be taken into account for correct consideration, as specified in equations 21, 23 and 25. This applies equally to the use of a one-dimensional, a two-dimensional and a three-dimensional heat equation and equally to any type of heat equation. As also mentioned earlier, in connection with solving the heat conduction equation In each case, a phase transformation equation can also be solved if this is necessary.

In principle, the technical application is always possible if the temperature of the rolling stock 1 is to be modeled. Examples of such matters have been given above in connection with 3 explained in detail. However, other applications are also possible.

It is well known to those skilled in the art that the heat conduction equation according to Equation 8 (this also applies analogously to other heat conduction equations) can be solved in real time. In this case, the modeling of the development over time of the thermal state Z of the rolling stock 1 takes place in real time. For example, in connection with the 3 and 4 explained procedures this is the case.

In a similar way, an online solution is also possible, i.e. not in real time, but in close temporal coupling with a real process. For example, as part of a setup calculation, control device 11 can be supplied with an expected initial thermal state ZA and an expected course over time for the speed v of rolling stock 1, so that control device 11 can determine in advance which current thermal state Z is expected when the Rolling stock 1 reaches the treatment device 2.

The requirements for online execution are lower than the requirements for real-time execution. Since it was explained above how real-time execution can be implemented, online execution is also possible.

The present invention has many advantages. In particular, the influence of the density ρ can be taken into account at least substantially correctly in the heat conduction equation even if the density ρ is state-dependent. This is an improved modeling of the thermal Behavior of the rolling stock 1 possible. On the other hand, it is not necessary to shift the support points for which the heat conduction equation is solved.

Although the invention has been illustrated and described in detail by the preferred embodiment, the invention is not limited by the disclosed examples and other variants can be derived therefrom by a person skilled in the art without departing from the protective scope of the invention.

Reference List

1

rolling stock

2

treatment facility

3

roughing stand

4

finishing stand

5

application devices

6, 8

heating devices

7, 9

descaling equipment

10

interstand cooling

11

control device

12

control program

13

machine code

14

sections

15, 17

measuring devices

16

model

18

volume element

A

control

a, a'

factors

B, B0

latitudes

D, D0, ds, ds0

thick

H, H0

Enthalpies or energy densities

j, j0

heat flux densities

L, L0

lengths

p

phase state

s

Position variable in thickness direction

S1 to S23

steps

T, T0

temperatures

t

time

v

speed

xL, xW, xD

coefficients

Z,ZA,ZT,Z0

thermal conditions

λ, λ0

thermal conductivities

p, p0

dense

Claims (8)

Treatment method for a rolled stock (1) made of metal, wherein the rolled stock (1) is a flat hot rolled stock extending in a longitudinal direction, in a width direction and in a thickness direction, - the rolling stock (1) being acted on at least essentially in the thickness direction at a treatment time by means of a treatment device (2), - the development over time of a thermal state (Z) of the rolling stock (1) being modeled at least for a period before the treatment time by means of a model (16) of the rolling stock (1) by iteratively solving at least one heat conduction equation, - the treatment device (2), on the basis of which the treatment device (2) acts on the rolling stock (1), being controlled as a function of that thermal state (Z) which is determined by the model (16) for the rolling stock (1) for the treatment time is determined, - where the density (ρ) of the rolling stock (1) is included in the heat conduction equation, - where the density (ρ) depends on the respective thermal state (Z) of the rolling stock (1), characterized,
that the dependence of the density (p) of the rolling stock (1) on the respective thermal state (Z) of the rolling stock (1) in the heat conduction equation by a factor (a, a') of the form $$a=\frac{\mathit{xL}\cdot \mathit{xB}}{\mathit{xD}\cdot {\rho }_0}\mathrm{or}\mathit{aʹ}=\frac{1}{{\mathit{xD}}^2\cdot \rho }$$

is taken into account, where ρ and ρ ₀ are densities (ρ, ρ ₀ ) of the rolling stock (1) related to a current and a predetermined thermal state (Z, Z0) of the rolling stock (1) and the coefficients xL, xB and xD are as Quotient of an extension (L, B, D, L ₀ , B ₀ , D ₀ ) of the rolling stock (1) in the longitudinal direction, in the width direction and in the thickness direction for the respective and the predetermined thermal state (Z, Z0) of the rolling stock (1). Treatment method according to claim 1,
characterized,
that the coefficients xL, xB and xD are determined in such a way that they are equal to each other and their product is equal to the quotient of the density (ρ) dependent on the respective thermal state (Z) of the rolling stock (1) and the predetermined thermal state ( Z0) of the rolling stock (1) related normalized density (ρ ₀ ) or the coefficient xD is determined in such a way that it is equal to the third root of the quotient of the respective thermal state (Z) of the rolling stock (1) dependent density (ρ) and is the normalized density (ρ ₀ ) related to the predetermined thermal state (Z0) of the rolling stock (1). Treatment method according to claim 1 or 2,
characterized,
that the modeling of the temporal development of the thermal state (Z) of the rolling stock (1) takes place online or in real time. Treatment method according to claim 1, 2 or 3,
characterized,
that the rolling stock (1) is rolled by means of the treatment device (2), so that the thickness (D) of the rolling stock (1) after the action of the treatment device (2) on the rolling stock (1) is smaller than before the action of the treatment device ( 2) on the rolling stock (1). Treatment method according to claim 1, 2 or 3,
characterized,
that the treatment device (2) is used to influence the rolling stock (1) purely thermally without forming the rolling stock (1), in particular heating or cooling the rolling stock (1). Control program for a control device (11) of a treatment device (2) for treating a rolling stock (1) made of metal, the control program comprising machine code (13) which can be processed by the control device (11), the processing of the machine code (13) being carried out by the control device (11) causes the control device (11) to operate the treatment device (2) according to a treatment method according to any one of the above claims. Control device of a treatment device (2) for treating a rolling stock (1) made of metal, wherein the control device is programmed with a control program (12) according to Claim 6, so that the control device controls the treatment device (2) according to a treatment method according to one of Claims 1 to 5 operates. Treatment plant for treating a rolled stock (1) made of metal, the rolled stock (1) being a flat hot rolled stock extending in a longitudinal direction, in a width direction and in a thickness direction, - the treatment plant having a treatment device (2), by means of which the rolling stock (1) can be acted on at least essentially in the thickness direction, - wherein the treatment system has a control device (11) by which at least the treatment device (2) is controlled, - Wherein the control device (11) is designed as a control device according to claim 7, so that the control device (11) operates the treatment device (2) according to a treatment method according to one of claims 1 to 5.

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