EP1397523B1 - Cooling method for a hot-rolled product and a corresponding cooling-section model - Google Patents

Abstract

To determine the temperature profile (Tm(t)) of a hot-rolled material (1) in a cooling line (5), a heat conduction equation which takes the following formwhere e is the enthalpy, lambda the thermal conductivity, p the degree of phase transformation, rho the density and T the temperature of the rolled material at the rolled-material location and t is the time, is solved in a cooling-line model (4).

Description

• vor der Kühlstrecke wird für eine Walzgutstelle eine Anfangstemperatur erfasst,
• anhand eines Kühlstreckenmodells und vorgegebener Solleigenschaften des Walzgutes wird ein zeitlicher Kühlmittelmengenverlauf ermittelt,
• auf die Walzgutstelle wird gemäß dem ermittelten zeitlichen Kühlmittelmengenverlauf ein Kühlmittel aufgebracht, und
• anhand des Kühlstreckenmodells und des zeitlichen Kühlmittelmengenverlaufs wird ein erwarteter zeitlicher Temperaturverlauf des Walzgutes an der Walzgutstelle über den Walzgutquerschnitt ermittelt.

The present invention relates to a cooling method for a hot-rolled rolling stock having a Walzgutquerschnitt, in particular a metal strip, for. As a steel strip, in a cooling section, with the following steps:

• in front of the cooling section, an initial temperature is detected for a rolling stock
• Based on a cooling line model and predetermined desired properties of the rolling stock, a time course of coolant quantity is determined,
• on the Walzgutstelle a coolant is applied in accordance with the determined temporal coolant flow rate, and
• Based on the cooling section model and the temporal coolant flow rate an expected temporal temperature profile of the rolling stock is determined at the Walzgutstelle on the Walzgutquerschnitt.

The present invention further relates to a cooling line model corresponding thereto.

Such a cooling method and the corresponding cooling line model are z. B. " Stahl und Eisen ", Vol. 116 (1996), No. 11, pages 115 to 120 known.

When cooling a hot-rolled metal strip, the exact modeling of the temporal temperature profile is crucial for the control of the coolant flow rate. Further, since the cooling does not take place in the thermodynamic equilibrium, affect phase transitions of the rolling stock to be cooled, for. As a phase transformation of steel, crucial to the thermal behavior during cooling. The phase transformation must therefore be included in the Fourier heat equation.

The modeling of the phase transformation again requires the temperature as input parameter. This creates a coupled differential equation system, the numerically z. B. can be solved by an initial value problem solver approximately. In this approach, the Fourier heat equation has to be solved together with the dynamics of the phase transformation.

Two methods are common in the prior art.

In the first, the phase transformation is first modeled based on an approximate temperature history. Thereafter, the phase transformation is frozen. The exothermic processes in the phase transformation are then taken into account by heat sources in the Fourier heat equation. This approach partially neglects the coupling between the phase transformation and the temperature.

In another method, although the Fourier heat equation is solved coupled with the phase transformation. Also in this method, exothermic processes in the phase transformation by heat sources are modeled in the Fourier heat equation.

By the methods of the prior art, the problem is only apparently solved. Because the approach is physically wrong in both cases. This is particularly evident in the fact that the heat source in the cooling line model must be parameterized separately.

The object of the present invention is to provide a cooling method and the corresponding cooling line model, by means of which the temperature of the rolling stock to be cooled as well as its phases and phase transitions are described correctly.

gelöst wird, wobei e die Enthalpie, λ die Wärmeleitfähigkeit, p der Phasenumwandlungsgrad, ρ die Dichte und T die Temperatur des Walzgutes an der Walzgutstelle und t die Zeit ist.The object is achieved for the cooling method in that for determining the temperature profile in the rolling stock in the cooling section model, a heat equation of the form $$\frac{\mathit{\partial }\mathit{e}}{\mathit{\partial }\mathit{t}}\mathit{-}\mathit{div}\left[\frac{\mathit{\lambda }\left(\mathit{e},,\mathit{p}\right)}{\mathit{\rho }}\mathit{\cdot }\mathit{gradT}\left(\mathit{e},,\mathit{p}\right)\right]=0$$

where e is the enthalpy, λ is the thermal conductivity, p is the degree of phase transformation, ρ is the density, and T is the temperature of the rolling stock at the rolling stock and t is the time.

The quantities e and p are location and time dependent. div and degrees are the well-known operators divergence and gradient, which act on the place variables.

enthält, wobei e die Enthalpie, λ die Wärmeleitfähigkeit, p der Phasenumwandlungsgrad, ρ die Dichte und T die Temperatur des Walzgutes an der Walzgutstelle und t die Zeit ist.Correspondingly, the task for the cooling section model is achieved in that, to determine the temperature profile in the rolling stock, a heat equation of the shape $$\frac{\mathit{\partial }\mathit{e}}{\mathit{\partial }\mathit{t}}\mathit{-}\mathit{div}\left[\frac{\mathit{\lambda }\left(\mathit{e},,\mathit{p}\right)}{\mathit{\rho }}\mathit{\cdot }\mathit{gradT}\left(\mathit{e},,\mathit{p}\right)\right]=0$$

where e is the enthalpy, λ is the thermal conductivity, p is the degree of phase change, ρ is the density and T is the temperature of the rolling stock at the rolling stock and t is the time.

The above equation is still in the usual form to complement initial and boundary conditions. These additions are made in the same way as well as in the prior art, common practice and known. The additions will therefore not be discussed further below.

The approach according to the invention is based on the principle of energy conservation. The Fourier heat conduction line is therefore formulated with the enthalpy as a state variable and the temperature as the size dependent on the enthalpy. heat sources are obviously not needed. They do not have to be parameterized any more.

Due to the now correct approach for the heat equation, the phase transformation degree and the enthalpy represent state variables that can be numerically calculated in parallel.

vereinfacht werden. x bezeichnet dabei die Ortsvariable in Banddickenrichtung.The above solution is independent of the profile of the rolling stock to be cooled. If the rolling stock is a metal strip, essentially results in a heat flow only in the direction of the strip thickness. In the strip running direction and in the bandwidth direction, however, only a negligible heat flow occurs. It is therefore possible to reduce the computational effort by the fact that the heat equation is considered only one-dimensional instead of three-dimensional. In this case, therefore, the heat equation can $$\frac{\mathit{\partial }\mathit{e}}{\mathit{\partial }\mathit{t}}\mathit{-}\frac{\mathit{\partial }}{\mathit{\partial }\mathit{x}}\left[\frac{\mathit{\lambda }\left(\mathit{e},,\mathit{p}\right)}{\mathit{\rho }}\mathit{\cdot }\frac{\mathit{\partial }T\mathit{(}\mathit{e}\mathit{.}\mathit{p}\mathit{)}}{\mathit{\partial }\mathit{x}}\right]=0$$

be simplified. x denotes the position variable in the band thickness direction.

The modeling is even better if a final temperature is recorded for the rolling stock behind the cooling section. Because then it is in particular possible to adapt the cooling line model based on a comparison of the detected end temperature with a determined based on the expected temporal temperature profile expected end temperature. Thus, the model can be optimized based on the actual detected final temperature.

ermittelt wird. Der Vorteil diese Ansatzes besteht in der Möglichkeit der Kopplung an die Fouriersche Wärmeleitungsgleichung, ohne dass dabei die Möglichkeit aufgegeben werden muss, einen Anfangswertproblemlöser zur gekoppelten Berechnung von Phasenumwandlungsgrad p und Temperatur T einzusetzen.As part of the cooling line model, it is necessary to also determine the phase transformation degree. This can be done in different ways. For example, it is possible to have the phase conversion degree according to Scheil's rule to investigate. It is also possible, for example, for the phase transformation degree (p) in the cooling-gap model to be based on a differential equation of the shape $$\frac{\mathit{\partial }\mathit{p}}{\mathit{\partial }\mathit{t}}=H\left(e,,p\right)$$

is determined. The advantage of this approach is the possibility of coupling to the Fourier heat equation, without sacrificing the possibility of using an initial value problem solver for the coupled calculation of phase transformation degree p and temperature T.

h is a function such as In equation 2 on page 144 of the article " Mathematical Models of Solid-Solid Phase Transitions in Steel "by A. Visintin, IMA Journal of Applied Mathematics, 39, 1987, pages 143-157 is disclosed.

FIG 1

eine Kühlstrecke mit einem Metallband,

FIG 2

ein Kühlstreckenmodell,

FIG 3

die Wärmeleitfähigkeit als Funktion der Enthalpie für zwei verschiedene Phasenumwandlungsgrade,

FIG 4

die Temperatur als Funktion der Enthalpie für zwei verschiedene Phasenumwandlungsgrade und

FIG 5

ein Wärmeleitungsmodell.

Further advantages and details emerge from the following description of an embodiment in conjunction with the drawings. This show in a schematic representation

FIG. 1

a cooling section with a metal band,

FIG. 2

a cooling line model,

FIG. 3

the thermal conductivity as a function of enthalpy for two different phase transformation degrees,

FIG. 4

the temperature as a function of enthalpy for two different phase transformation degrees and

FIG. 5

a heat conduction model.

According to FIG. 1, a hot-rolled rolling stock 1 runs out of a rolling stand 2 at a rolling speed v in a strip running direction z. Behind the rolling stand 2, a rolling stand temperature measuring station 3 is arranged. In rolling mill temperature measuring station 3, an initial temperature T1 is determined for a rolling stock detected on the surface of the rolling stock 1 and fed to a cooling line model 4 as an input parameter.

According to FIG 1, the rolling stock 1 is a metal strip, for. B. a steel strip. It therefore has a rolling stock width b in a width direction y and a rolling stock thickness d in a thickness direction x. Walzgutbreite b and Walzgutdicke d together give the Walzgutquerschnitt the rolling stock. 1

The initial temperature T1 of the rolling stock 1 can vary across the bandwidth b. The rolling temperature measuring station 3 is therefore preferably designed such that the initial temperature T1 across the bandwidth b can be detected multiple times. For example, a plurality of temperature sensors arranged transversely across the bandwidth b may be provided for this purpose. It is also possible to provide a temperature sensor, which is preceded by an optical system, by means of which in the bandwidth direction y scanning is possible.

Behind the rolling stand temperature measuring station 3, a cooling section 5 is arranged. The cooling section 5 has cooling devices 6, by means of which a coolant 7, typically water 7, from above, from below or from both sides of the rolling stock 1 can be applied. The type of application is adapted to the profile to be rolled.

Behind the cooling section 5, a reel temperature measuring station 8 is arranged. With this a corresponding end temperature T2 can be detected for the Walzgutstelle, which is also supplied to the cooling line model 4. The reel temperature measuring station 8 is designed in the same way as the rolling stand temperature measuring station 3.

The reel temperature measuring station 8 is followed by a reel 9. On this, the metal strip 1 is reeled.

The arrangement of the reel 9 is typical when rolling tapes. When rolling profiles, instead of the reel 9 usually another unit is provided, for. B. in wire rod mills a Windungsleger.

The rolling stock 1 should have a predetermined temperature and desired desired microstructural properties G * when the reel 9 is reached. For this purpose, it is necessary that the metal strip 1 between rolling stand 2 and reel 9 has a corresponding temperature profile. This temperature profile is calculated by means of the cooling section model 4.

According to FIGS. 1 and 2, different values are supplied to the cooling line model 4. First, the cooling speed model 4, the rolling speed v is supplied. Due to this fact, in particular a material tracking is feasible.

Then, the cooling section model 4, the strip thickness d, the initial temperature T1 and various parameters PAR supplied. The parameters PAR include in particular actual and desired parameters of the strip 1. An actual parameter is, for example, the alloy of the metal strip 1 or its bandwidth b. A desired parameter is, for example, the desired reel temperature.

The cooling line model 4 according to FIG. 2 comprises a heat conduction model 10, a heat transfer model 11 and a coolant quantity course determiner 12. The cooling stretch model 4 then determines an expected temporal temperature profile Tm (t). The expected temperature profile Tm (t) is compared with a desired temperature profile T * (t). The comparison result is supplied to the coolant quantity course determiner 12. This then uses the difference to determine a new coolant flow rate in order to approximate the expected temperature curve Tm (t) to the desired temperature curve T * (t).

After the adjustment, the cooling devices 6 of the cooling section 5 are then controlled by the Kühlmengenverlaufsermittler 12 accordingly. The coolant 7 is thus applied to the relevant Walzgutstelle according to the determined temporal coolant flow rate.

auf. In der Formel bezeichnen e die Enthalpie, λ die Wärmeleitfähigkeit, p den Phasenumwandlungsgrad, ρ die Dichte und T die Temperatur des Walzgutes 1 an der Walzgutstelle sowie t die Zeit.In order to determine the expected temperature profile Tm (t), a heat conduction equation is achieved in the heat conduction model 10. The heat equation has the shape $$\frac{\mathit{\partial }\mathit{e}}{\mathit{\partial }\mathit{t}}\mathit{-}\mathit{div}\left[\frac{\mathit{\lambda }\left(\mathit{e},,\mathit{p}\right)}{\mathit{\rho }}\mathit{\cdot }\mathit{gradT}\left(\mathit{e},,\mathit{p}\right)\right]=0$$

on. In the formula, e denotes the enthalpy, λ the thermal conductivity, p the phase conversion degree, ρ the density and T the temperature of the rolling stock 1 at the rolling stock and t the time.

h ist eine Funktion wie sie z. B. in Gleichung 2 auf Seite 144 des Artikels " Mathematical Models of Solid-Solid Phase Transitions in Steel" von A. Visintin, IMA Journal of Applied Mathematics, 39, 1987, Seiten 143 bis 157 offenbart ist.For the correct solution of the heat conduction equation, the phase transformation degree p and its time course must also be determined. This is preferably done using a differential equation of the form $$\frac{\mathit{\partial }\mathit{p}}{\mathit{\partial }\mathit{t}}=H\left(e,,p\right)$$

h is a function such as In equation 2 on page 144 of the article " Mathematical Models of Solid-Solid Phase Transitions in Steel "by A. Visintin, IMA Journal of Applied Mathematics, 39, 1987, pages 143-157 is disclosed.

The above equations must be solved at the Walzgutstelle for the entire Walzgutquerschnitt. Furthermore, where appropriate, the heat flow in the direction of strip z must be taken into account.

angenähert werden. Dabei sind in beispielhafter Ausgestaltung λ(e,1) und λ (e,0) Funktionen wie sie in FIG 3 gezeigt sind.The relationship λ (e, p) can be found in the equations z. B. by the function $$\mathit{\lambda }\left(\mathit{e},,\mathit{p}\right)\mathit{=}\mathit{p\lambda }\left(e,,1\right)+\left(1-p\right)\mathit{\lambda }\left(e,,0\right)$$

be approximated. In an exemplary embodiment, λ (e, 1) and λ (e, 0) are functions as shown in FIG.

angenähert werden. Dabei sind T(e,1) und T(e,0) Funktionen wie sie beispielhaft in FIG 4 gezeigt sind.The relationship T (e, p) can z. B. by the function $$\mathit{T}\left(\mathit{e},,\mathit{p}\right)\mathit{=}\mathit{pT}\left(e,,1\right)+\left(1-p\right)\mathit{T}\left(e,,0\right)$$

be approximated. Here, T (e, 1) and T (e, 0) are functions as shown by way of example in FIG.

As long as the metal strip 1 has not yet reached the reel temperature measuring station 8, only the initial temperature T1 is available as the actual temperature value. On the other hand, as soon as the end temperature T2 can also be detected, it can be compared with an end temperature T2m expected on the basis of the previous calculation. The comparison result is fed to an adaptation element 13. By means of the adaptation element 13, for example, the heat transfer model 13 can be adapted.

gelöst. Beim Kühlen von Metallband erfolgt ein Wärmefluss aber im wesentlichen ausschließlich in x-Richtung. Es ist daher möglich und zulässig, gemäß FIG 5 das Wärmeleitungsmodell 10 eindimensional anzusetzen. Es ist also hinreichend eine Wärmeleitungsgleichung der Form $$\frac{\mathit{\partial }\mathit{e}}{\mathit{\partial }\mathit{t}}\mathit{-}\frac{\mathit{\partial }}{\mathit{\partial }\mathit{x}}\left[\frac{\mathit{\lambda }\left(\mathit{e},,\mathit{p}\right)}{\mathit{\rho }}\mathit{\cdot }\frac{\mathit{\partial }T\mathit{(}\mathit{e}\mathit{,}\mathit{p}\mathit{)}}{\mathit{\partial }\mathit{x}}\right]=0$$

zu lösen. Diese Vorgehensweise erfordert einen erheblich geringeren Rechenaufwand bei nur geringfügig verschlechterten Ergebnis, weil in diesem Fall lediglich die Wärmeleitungsgleichung für einen eindimensionalen Stab, der sich an der Walzgutstelle von der Bandunterseite zur Bandoberseite erstreckt, gelöst werden muss.In the illustrated in Figure 2 and above explained cooling section model 4 is in the context of the heat conduction model 10, the heat equation $$\frac{\mathit{\partial }\mathit{e}}{\mathit{\partial }\mathit{t}}\mathit{-}\mathit{div}\left[\frac{\mathit{\lambda }\left(\mathit{e},,\mathit{p}\right)}{\mathit{\rho }}\mathit{\cdot }\mathit{gradT}\left(\mathit{e},,\mathit{p}\right)\right]=0$$

solved. When cooling metal strip, however, a heat flow takes place essentially exclusively in the x direction. It is therefore possible and permissible, according to FIG 5, the heat conduction model 10 one-dimensionally set. So it is sufficient a heat equation of the form $$\frac{\mathit{\partial }\mathit{e}}{\mathit{\partial }\mathit{t}}\mathit{-}\frac{\mathit{\partial }}{\mathit{\partial }\mathit{x}}\left[\frac{\mathit{\lambda }\left(\mathit{e},,\mathit{p}\right)}{\mathit{\rho }}\mathit{\cdot }\frac{\mathit{\partial }T\mathit{(}\mathit{e}\mathit{.}\mathit{p}\mathit{)}}{\mathit{\partial }\mathit{x}}\right]=0$$

to solve. This approach requires a significantly lower computational effort with only slightly deteriorated result, because in this case, only the heat equation for a one-dimensional rod that extends at the Walzgutstelle from the belt bottom to the top of the band must be solved.

Claims (14)

1. Method for cooling a hot-rolled material (1) having a rolled-material cross section, in particular a metal strip (1), e.g. a steel strip (1), in a cooling line (5), comprising the following steps:

   - a starting temperature (T1) is recorded for a rolled-material location upstream of the cooling line (5),

   - a temporal quantitative coolant profile is determined on the basis of a cooling-line model (4) and predetermined desired properties of the rolled material (1),

   - a coolant (7) is applied to the rolled-material location in accordance with the temporal quantitative coolant profile which has been determined, and

   - an expected temporal temperature profile (Tm(t)) of the rolled material (1) at the rolled-material location across the rolled-material cross section is determined on the basis of the cooling-line model (4) and the temporal quantitative coolant profile,

   characterized
   in that a heat conduction equation of the following form $$\frac{\mathit{\partial }\mathit{e}}{\mathit{\partial }\mathit{t}}\mathit{-}\mathit{div}\left[\frac{\mathit{\lambda }\left(\mathit{e},,\mathit{p}\right)}{\mathrm{\rho }}\mathit{\cdot }\mathit{gradT}\left(\mathit{e},,\mathit{p}\right)\right]=0$$

   

   
   where e is the enthalpy, λ the thermal conductivity, p the degree of phase transformation, ρ the density and T the temperature of the rolled material at the rolled-material location and t is the time, is solved in the coolant-line model (4) in order to determine the temperature profile (Tm(t)) in the rolled material (1).
2. Cooling method according to claim 1, characterized
   in that a finishing temperature (T2) is recorded for the rolled-material location downstream of the cooling line (5).
3. Cooling method according to claim 2, characterized
   in that the cooling-line model (4) is adapted on the basis of a comparison between the recorded finishing temperature (T2) and an expected finishing temperature (T2m), which is determined on the basis of the expected temporal temperature profile (Tm(t)).
4. Method for cooling a hot-rolled metal strip (1), in particular a steel strip (1), having a strip thickness (d), in a cooling line (5), comprising the following steps:

   - a starting temperature (T1) is recorded for a strip location upstream of the cooling line (5),

   - a temporal quantitative coolant profile is determined on the basis of a cooling-line model (4) and predetermined desired properties of the metal strip (1),

   - a coolant (7) is applied to the strip location in accordance with the temporal quantitative coolant profile which has been determined, and

   - an expected temporal temperature profile (Tm(t)) of the metal strip (1) at the strip location across the strip thickness (d) is determined on the basis of the cooling-line model (4) and the temporal quantitative coolant profile,

   characterized in that a heat conduction equation of the following form $$\frac{\mathit{\partial }\mathit{e}}{\mathit{\partial }\mathit{t}}\mathit{-}\frac{\mathit{\partial }}{\mathit{\partial }\mathit{x}}\left[\frac{\mathit{\lambda }\left(\mathit{e},,\mathit{p}\right)}{\mathrm{\rho }}\mathit{\cdot }\frac{\mathit{\partial }T\mathit{(}\mathit{e}\mathit{,}\mathit{p}\mathit{)}}{\mathit{\partial }\mathit{x}}\right]=0$$

   

   
   where e is the enthalpy, x the position in the strip thickness direction, λ the thermal conductivity, p the degree of phase transition, ρ the density and T the temperature of the metal strip (1) at the strip location and t is the time, is solved in the cooling-line model (4) in order to determine the temperature profile (Tm(t)) in the metal strip (1).
5. Cooling method according to claim 4, characterized in that a finishing temperature (T2) is recorded for the strip location downstream of the cooling line (5).
6. Cooling method according to claim 5, characterized in that the cooling-line model (5) is adapted on the basis of a comparison between the recorded finishing temperature (T2) and an expected finishing temperature (T2m) which is determined on the basis of the expected temporal temperature profile (Tm(t)).
7. Cooling method according to one of the preceding claims, characterized in that the degree of phase transition (p) is determined in the cooling-line model (4) on the basis of a differential equation which takes the following form $$\frac{\mathit{\partial }\mathit{p}}{\mathit{\partial }\mathit{t}}=h\left(e,,p\right)\mathrm{.}$$

   
8. Cooling-line model for a hot-rolled material (1) which is to be cooled in a cooling line (5) and has a rolled-material cross section, in particular a metal strip (1), e.g. a steel strip (1), in which

   - a starting temperature (T1), recorded upstream of the cooling line (5), of a rolled-material location can be fed to the cooling-line model (4),

   - a temporal quantitative coolant profile can be determined by means of the cooling-line model (4) on the basis of predetermined desired properties of the rolled material (1),

   - an expected temporal temperature profile (Tm (t)) of the rolled material (1) at the rolled-material location across the rolled-material cross section can be determined by means of the cooling-line model (4) and the temporal quantitative coolant profile,

   characterized in that the cooling-line model (4), in order to determine the temperature profile (Tm(t)) in the rolled material (1), includes a heat conduction equation of the following form $$\frac{\mathit{\partial }\mathit{e}}{\mathit{\partial }\mathit{t}}\mathit{-}\mathit{div}\left[\frac{\mathit{\lambda }\left(\mathit{e},,\mathit{p}\right)}{\mathrm{\rho }}\mathit{\cdot }\mathit{gradT}\left(\mathit{e},,\mathit{p}\right)\right]=0$$

   

   
   where e is the enthalpy, λ the thermal conductivity, p the degree of phase transformation, ρ the density and T the temperature of the rolled material at the rolled-material location and t is the time.
9. Cooling-line model according to claim 8,
   characterized in that it can be fed a finishing temperature (T2), recorded downstream of the cooling line (5), of the rolled-material location.
10. Cooling method according to claim 9, characterized in that the cooling-line model (4) can be adapted on the basis of a comparison between the recorded finishing temperature (T2) and an expected finishing temperature (T2m), which is determined on the basis of the expected temporal temperature profile (Tm(t)).
11. Cooling-line model for a hot-rolled metal strip (1) which is to be cooled in a cooling line (5) and has a strip thickness (d), in particular a steel strip (1), in which

    - a starting temperature (T1), recorded downstream of the cooling line (5), of a strip location can be fed to the cooling-line model (4),

    - a temporal quantitative coolant profile can be determined by means of the cooling-line model (4) on the basis of predetermined desired properties of the metal strip (1),

    - an expected temporal temperature profile (Tm(t)) of the metal strip (1) at the strip location across the strip thickness (d) can be determined by means of the cooling-line model (4) and the temporal quantitative coolant profile,

    characterized in that the cooling-line model (4), in order to determine the temperature profile (Tm(t)) in the metal strip (1), includes a heat conduction equation which takes the following form $$\frac{\mathit{\partial }\mathit{e}}{\mathit{\partial }\mathit{t}}\mathit{-}\frac{\mathit{\partial }}{\mathit{\partial }\mathit{x}}\left[\frac{\mathit{\lambda }\left(\mathit{e},,\mathit{p}\right)}{\mathrm{\rho }}\mathit{\cdot }\frac{\mathit{\partial }T\mathit{(}\mathit{e}\mathit{,}\mathit{p}\mathit{)}}{\mathit{\partial }\mathit{x}}\right]=0$$

    

    
    where e is the enthalpy, x the position in the strip thickness direction, λ the thermal conductivity, p the degree of phase transformation, ρ the density and T the temperature of the metal strip (1) at the strip location and t is the time.
12. Cooling-line model according to claim 11, characterized in that it can be fed a finishing temperature (T2), recorded downstream of the cooling line (5), of the strip location.
13. Cooling method according to claim 12, characterized in that the cooling-line model (4) can be adapted on the basis of a comparison between the recorded finishing temperature (T2) and an expected finishing temperature (T2m) which has been determined on the basis of the expected temporal temperature profile (Tm(t)).
14. Cooling-line model according to one of claims 8 to 13,
    characterized in that to determine the degree of phase transformation (p) it includes a differential equation which takes the following form $$\frac{\mathit{\partial }\mathit{p}}{\mathit{\partial }\mathit{t}}=h\left(e,,p\right)\mathrm{.}$$

    

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