JP7560733B2 - Estimation device, method, and program, and continuous casting control device and method - Google Patents

Description

The present invention relates to an estimation device, method, and program, as well as a continuous casting control device and method, and is particularly suitable for use in casting a slab using a continuous casting machine.

In the continuous casting process, molten steel is supplied from a tundish into a mold, and the slab is drawn out from the mold. In a secondary cooling zone located downstream of the mold, cooling water is sprayed onto the slab to cool it, and then the slab is bent and straightened downstream of the secondary cooling zone. The slab is then cut into slabs, billets, blooms, and the like. Here, the exit of the roll located at the most downstream position in the drawing direction of the slab is called the end exit. If the cooling in the secondary cooling zone is not appropriate, there is a risk that cracks will occur on the surface of the slab, or that central segregation will occur in the slab. In addition, if solidification does not complete to the center of the slab at the end exit, and solidification of the slab completes beyond the end exit, there is a risk that the slab will expand significantly. Therefore, the temperature and solid fraction of the slab being cast in the continuous casting machine are estimated and managed (see, for example, Patent Documents 1 and 2).

The heat transfer coefficient between the slab and the outside is used to estimate the temperature and solid fraction of the slab. Therefore, if the accuracy of the heat transfer coefficient estimation is low, the accuracy of the estimation of the temperature and solid fraction of the slab will also be low. Therefore, Patent Documents 1 and 2 disclose a method of calculating the heat transfer coefficient for each of multiple cooling means applied to the slab, and performing a heat transfer calculation using the maximum value of the calculated heat transfer coefficients. Patent Document 2 also discloses a method of calculating the total heat transfer amount within each calculation interval at the position where the temperature of the slab is estimated, and performing a heat transfer calculation using the calculated total heat transfer amount.

JP 2013-35011 A JP 2019-98388 A

However, in the technology described in Patent Documents 1 and 2, the heat transfer calculation is performed at the downstream end of the calculation interval using the surface temperature of the slab at the upstream end of the calculation interval. Therefore, it is not taken into consideration that the surface temperature of the slab changes with casting within the same calculation interval. Therefore, even if the surface temperature of the slab changes within the same calculation interval, one heat transfer coefficient is selected for the calculation interval. For this reason, for example, if the heat transfer coefficient varies greatly depending on the surface temperature of the slab within the same calculation interval, there is a risk that the estimation accuracy of the temperature and solid fraction of the slab will decrease. In this case, it is possible to reduce the change in the heat transfer coefficient within the same calculation interval by shortening the interval between calculation intervals. However, doing so increases the calculation load, and there is a risk that the calculation will not be completed while casting progresses for the length of the calculation interval.

The present invention was made in consideration of the above problems, and aims to improve the accuracy of estimating the state of the slab cast by a continuous casting machine while minimizing the increase in the calculation load.

The estimation device of the present invention is an estimation device that estimates the state of a slab cast by a continuous casting machine, and includes a heat transfer and solidification calculation unit that performs heat transfer and solidification calculation for a tracking surface, which is a cross section of the slab perpendicular to the casting direction, at a plurality of heat transfer and solidification calculation positions that are set at intervals in the casting direction, and a heat transfer calculation unit that performs heat transfer calculation for the surface of the slab at at least one intermediate calculation position that is set between two of the heat transfer and solidification calculation positions that are adjacent to each other in the casting direction.
the heat transfer calculation unit executes a heat transfer calculation for the surface of the slab at the intermediate calculation position using a result of the heat transfer calculation for the tracking surface at an upstream heat transfer solidification calculation position of the two heat transfer solidification calculation positions adjacent to each other in the casting direction, and the heat transfer solidification calculation unit executes a heat transfer calculation for the tracking surface when the tracking surface at the upstream heat transfer solidification calculation position is located at a downstream heat transfer solidification calculation position of the two heat transfer solidification calculation positions adjacent to each other in the casting direction, using the result of the heat transfer calculation for the surface of the slab at the intermediate calculation position.

The continuous casting control device of the present invention includes the estimation device and a control device that controls the amount of cooling water supplied to the secondary cooling zone of the continuous casting machine so that the deviation between the target value and the calculated value of the temperature at a specified position of the slab being cast by the continuous casting machine approaches zero based on the results of the heat transfer and solidification calculation for the tracking surface performed by the estimation device.

The estimation method of the present invention is a method for estimating the state of a slab cast by a continuous casting machine, and includes a heat transfer solidification calculation step of performing a heat transfer solidification calculation for a tracking surface, which is a cross section of the slab perpendicular to the casting direction, at a plurality of heat transfer solidification calculation positions set at intervals in the casting direction, and a heat transfer calculation step of performing a heat transfer calculation for the surface of the slab at at least one intermediate calculation position set between two adjacent heat transfer solidification calculation positions in the casting direction, the heat transfer calculation step uses the result of the heat transfer calculation for the tracking surface at the upstream heat transfer solidification calculation position of the two adjacent heat transfer solidification calculation positions in the casting direction to perform a heat transfer calculation for the surface of the slab at the intermediate calculation position, and the heat transfer solidification calculation step uses the result of the heat transfer calculation for the surface of the slab at the intermediate calculation position to perform a heat transfer solidification calculation for the tracking surface when the tracking surface at the upstream heat transfer solidification calculation position is located at the downstream heat transfer solidification calculation position of the two adjacent heat transfer solidification calculation positions in the casting direction.

The continuous casting control method of the present invention includes a control step of controlling the amount of cooling water supplied to the secondary cooling zone of the continuous casting machine based on the results of the heat transfer and solidification calculation for the tracking surface performed by the estimation method, so that the deviation between the target value and the calculated value of the temperature at a specified position of the slab being cast by the continuous casting machine approaches zero.

The program of the present invention is intended to cause a computer to function as each of the means of the estimation device.

The present invention makes it possible to improve the accuracy of estimating the state of a slab cast by a continuous casting machine while minimizing the increase in the calculation load.

FIG. 1 is a diagram showing an example of a configuration of a continuous casting machine. FIG. 2 is a diagram illustrating an example of a cooling method for a cast slab. FIG. 13 is a diagram showing an example of calculated values of heat transfer coefficients in each region. FIG. 2 is a diagram illustrating an example of a functional configuration of an estimation device and a control device. FIG. 13 is a diagram conceptually illustrating an example of calculation points on a tracking surface of a heat transfer solidification calculation position. FIG. 13 is a diagram showing an example of a relationship between a heat transfer solidification calculation position and a temperature calculation position. FIG. 13 is a diagram conceptually illustrating an example of calculation points of a cross section subject to heat transfer calculation at an intermediate calculation position. 2 is a flowchart illustrating an example of a continuous casting control method. FIG. 1 is a diagram showing the relationship between the width center temperature and the casting length in a reference example and a comparative example. FIG. 1 is a diagram showing the relationship between the width center temperature and the casting length in a reference example and an inventive example. FIG. 11 is an enlarged view of a portion of FIG. 10 .

Hereinafter, an embodiment of the present invention will be described with reference to the drawings.
In addition, the term "compared objects" including length, position, size, spacing, etc., being the same includes not only objects that are strictly the same, but also objects that are different within the scope of the invention (for example, objects that differ within the tolerance range determined at the time of design).

(Outline of the configuration of the continuous casting machine)
Fig. 1 is a diagram showing an example of the configuration of a continuous casting machine. Note that since the continuous casting machine itself can be realized by known technology, Fig. 1 shows only the parts necessary for explaining the present embodiment in a simplified manner.

Molten steel is supplied into the mold 1 from a tundish (not shown) through an immersion nozzle (not shown) or the like. A molten steel meniscus 4 is formed at the top of the molten steel in the mold 1. The molten steel is cooled in the mold 1, and the slab 5, the outer side of which has solidified, is supported by a plurality of support rolls 7a to 7v, 8a to 8p and pulled out of the mold 1 at a drawing speed (casting speed). The support rolls 7a to 7v, 8a to 8p are arranged in opposing positions to sandwich the slab 5 therebetween, and are arranged at a predetermined interval in the casting direction. Between two support rolls adjacent in the casting direction, cooling sprays 2a to 2t are arranged to spray cooling water toward the slab 5. In this embodiment, the cooling sprays 2a to 2t are mist sprays that spray mist-like cooling water (a gas-liquid two-phase fluid of water and air) as an example. However, the cooling water that cools the slab 5 is not limited to mist-like cooling water. For example, a water spray (a spray that ejects a single-phase fluid consisting of water only) may be used instead of or in addition to a mist spray.

The flow rate of the cooling water sprayed by the cooling sprays 2a to 2t is adjusted by the flow rate control valves 3a to 3e installed in the pipes 10a to 10j. The opening degree of the flow rate control valves 3a to 3e is adjusted based on the water volume indication value given by the control device 100b. In this embodiment, the case where the shape of the cross section of the slab 5 in the direction perpendicular to the casting direction is rectangular will be described as an example. In the following description, the surface (5b, 5c) of the slab 5 that is the upper surface facing upward or the lower surface facing downward (in the direction of gravity) on the downstream side of the casting direction (near the machine end outlet 9) is referred to as the long side surface of the slab 5, and the surface (5a) of the slab 5 that is the side surface facing the horizontal direction is referred to as the short side surface of the slab 5. In this embodiment, the case where the cooling sprays 2a to 2t are arranged at positions facing the long side surfaces 5b to 5c of the slab 5 without arranging the cooling sprays at positions facing the short side surface 5a of the slab 5 will be described as an example. Therefore, the cooling sprays 2a to 2t spray cooling water onto the long side surfaces 5b to 5c of the slab 5. In this embodiment, the continuous casting control device 100 includes an estimation device 100a and a control device 100b as independent devices, and the flow rate of the cooling water in the cooling sprays 2a to 2t (hereinafter referred to as the cooling water amount) is controlled by the control device 100b based on the surface and internal temperatures of the slab 5 estimated by the estimation device 100a.

Pipes 10a to 10j are installed to correspond to multiple cooling zones (cooling zones divided by cooling zone boundary lines 6a to 6f) that divide the length of the strand in the casting direction. The distribution of the amount of cooling water in the casting direction within the strand (inside the machine) is controlled for each cooling zone. The entire set of these cooling zones (between cooling zone boundary lines 6a to 6f) is called the secondary cooling zone.

The support roll 8h also functions as a casting speed measurement roll for sequentially measuring the casting speed of the slab 5. In FIG. 1, an example is shown in which the lower support roll 8h of a pair of support rolls 8h, 8p arranged at the most downstream position in the casting direction (drawing direction) is used as the casting speed measurement roll. As explained in the Background section, the outlet of the pair of support rolls 8h, 8p arranged at the most downstream position in the casting direction becomes the machine end outlet 9.

(Heat transfer and solidification model)
In order to manage the operating state of the continuous casting machine, it is necessary to calculate the surface temperature, internal temperature, and solid phase ratio of the slab 5. In this embodiment, a heat transfer solidification calculation is performed to calculate the surface temperature, internal temperature, and solid phase ratio of the slab 5. An example of a heat transfer solidification model used in the heat transfer solidification calculation is described below. In this embodiment, a case where a heat transfer solidification calculation is performed using the heat transfer solidification models described in JP 2019-48322 A and JP 2019-141893 A is used as an example for description. Therefore, here, the main parts of the heat transfer solidification model will be described, and detailed descriptions will be omitted.

The cross section of the slab 5 cut in a direction perpendicular to the casting direction at heat transfer solidification calculation positions set at a constant interval Δz in the casting direction in the region from the molten steel surface (molten steel meniscus 4) in the mold 1 to the machine end outlet 9 is referred to as the heat transfer solidification calculation cross section. As described in JP 2019-48322 A and JP 2019-141893 A, the distribution of the temperature and solid fraction of the slab 5 in the strand is calculated by solving a heat conduction equation in which the distribution of the temperature and solid fraction of the slab 5 in each heat transfer solidification calculation cross section is discretized under the boundary condition of the heat transfer coefficient reflecting the cooling condition at each calculation point in each heat transfer solidification calculation cross section. Here, the heat transfer coefficient is the heat transfer coefficient between the slab 5 and the outside (this also applies to the following explanation).

In addition, in JP 2019-48322 A and JP 2019-141893 A, the heat transfer solidification calculation target cross section is referred to as the calculation target cross section. In this embodiment, in order to distinguish it from the heat transfer calculation target cross section described later, what is referred to as the calculation target cross section in JP 2019-48322 A and JP 2019-141893 A is referred to as the heat transfer solidification calculation target cross section. Note that, although details will be described later, the heat transfer calculation target cross section is set between two heat transfer solidification calculation cross sections adjacent in the casting direction in order to improve the accuracy of the heat transfer solidification calculation without shortening the interval between the two heat transfer solidification calculation target cross sections adjacent in the casting direction, and is not described in JP 2019-48322 A and JP 2019-141893 A.

As the initial condition of the heat conduction equation for each heat transfer solidification calculation cross section, the calculation results for the heat transfer solidification calculation cross section adjacent to the heat transfer solidification calculation cross section on the upstream side are set. When the slab 5 is pulled out, the calculations of the enthalpy, temperature (temperature distribution), and solid phase ratio within the heat transfer solidification calculation cross section are repeatedly calculated until the heat transfer solidification calculation cross section moves from the heat transfer solidification calculation position where the heat transfer solidification calculation cross section exists to the heat transfer solidification calculation position one position downstream of the heat transfer solidification calculation position. By performing such repeated calculations for each heat transfer solidification calculation cross section, the overall temperature and solid phase ratio of the slab 5 are calculated.

In order to respond to changes in the indicated values of the casting speed and the amount of cooling water, a new heat transfer solidification calculation target cross section is generated at the position of the molten steel meniscus 4 in the mold 1 (z = 0) each time the slab 5 is withdrawn by a certain distance Δz, and the position of the heat transfer solidification calculation target cross section is tracked sequentially. In the following description, this heat transfer solidification calculation target cross section is referred to as a tracking surface. When the tracking surface is located at the heat transfer solidification calculation position set at a certain interval Δz in the casting direction, a calculation is performed using a heat transfer solidification model for the tracking surface. On the tracking surface, calculation points for calculating the temperature and solid phase ratio are arranged in a lattice pattern. Calculation points are also arranged on the boundary line of the tracking surface. As described above, in this embodiment, the shape of the cross section in the direction perpendicular to the casting direction of the slab 5 is rectangular. Therefore, the shape of the tracking surface is also rectangular. The boundary line of the tracking surface is the outline of the rectangle and corresponds to the surface of the slab 5. In this way, the temperature and solid phase ratio of each calculation point arranged on the tracking surface are calculated based on the heat transfer solidification model. Calculations using the heat transfer solidification model are performed for each tracking surface located from the molten steel meniscus 4 to at least the outlet position of the secondary cooling zone.

In this embodiment, an example will be described in which the casting direction interval of the heat transfer solidification calculation positions and the casting direction interval for generating the tracking surfaces are both constant intervals Δz. This allows heat transfer solidification calculations to be performed for all tracking surfaces at the same time. As described in JP 2019-48322 A and JP 2019-141893 A, the casting direction interval for generating the tracking surfaces may be an integer multiple of the casting direction interval of the heat transfer solidification calculation positions. In other words, the integer multiple is not limited to 1, and may be an integer multiple of 2 or more.

As described above, the initial condition of the heat conduction equation for each tracking surface is set to the calculation result for the adjacent tracking surface on the upstream side of the tracking surface. By pulling out the slab 5, the calculation of the time change of the enthalpy, temperature, and solid fraction within the tracking surface until the tracking surface located at the heat transfer solidification calculation position moves from the heat transfer solidification calculation position to the heat transfer solidification calculation position one position downstream of the heat transfer solidification calculation position is repeated for each tracking surface, and the overall temperature and solid fraction of the slab 5 are calculated.

<Cross-section coordinates and state variables>
As described above, in this embodiment, the shape of the tracking surface is rectangular. Here, in a cross section perpendicular to the casting direction of the slab 5, the direction along the sides corresponding to the long side surfaces 5b-5c of the slab 5 is referred to as the width direction, and the direction along the short side surface 5a of the slab 5 is referred to as the thickness direction. In FIG. 1, the direction along the broken lines indicating the cooling zone boundaries 6a-6f is the thickness direction, the direction perpendicular to the paper surface of FIG. 1 is the width direction, and the direction perpendicular to the width direction and the thickness direction is the casting direction. The coordinates in the cross section perpendicular to the casting direction of the slab 5 are coordinates with a certain vertex of the slab 5 as the origin 0 (zero). The axis in the width direction of this coordinate is the x axis, and the axis in the thickness direction is the y axis. The width of the slab 5 is referred to as X, and the thickness is referred to as Y. The z axis of this coordinate is the axis in the drawing direction (i.e., the casting direction) of the slab 5. In FIG. 1, the position of the molten steel meniscus 4 is shown in x-y-z coordinates. As a symbol indicating x-y-z coordinates, a symbol with an x inside a circle corresponds to an arrow line pointing from the front to the back of the paper (the symbol in Figure 1 indicates that the direction from the front to the back of the paper is the positive direction of the x-axis).

In the heat transfer solidification model, the following are defined as state variables at coordinates (x, y) in a tracking plane (cross section for heat transfer solidification calculation) at time t: where μ is a number for identifying a solute component in molten steel.
Enthalpy: H(x, y, t)
Temperature: T (x, y, t)
Solid phase ratio: f _s (x, y, t)
Solute concentration on the liquid phase side at the solid-liquid interface: C _Lμ (x, y, t)
The physical property parameters in the heat transfer solidification model are as follows:
Thermal conductivity (temperature dependent): λ(x,y,t) = λ(T(x,y,t))
Specific heat of steel (temperature dependent): c(x,y,t) = c(T(x,y,t))
Latent heat of solidification: L _h
Liquidus temperature: T _L
Density: ρ

<Basic equations for heat transfer and solidification model>
Since enthalpy is the energy at each calculation point within the tracking surface, including the latent heat of solidification, its change over time can be expressed by the following equation (1), which is a heat conduction equation that represents the heat balance at each calculation point.

Here, _qx represents the heat flux in the x-axis direction, and _qy represents the heat flux in the y-axis direction. At inner points other than the boundary of the tracking surface, the following formulas (2) and (3) hold.

Here, λ _x represents the thermal conductivity in the x-axis direction, and λ _y represents the thermal conductivity in the y-axis direction. Therefore, from equations (2) and (3), equation (1) at an inner point of the tracking surface can be expressed as the following equation (4).

<Boundary conditions>
Among the positions on the boundary line of the tracking surface, at position x= _xB ( _xB =0 or X), the heat flux is expressed by the following formula (5), and at position y= _yB ( _yB =0 or Y), the heat fluxes _qx and _qy are expressed by the following formula (6). Note that x= _xB is the position in the width direction (x-axis coordinate) of the short side surface 5a of the slab 5, and y= _yB is the position in the thickness direction (y-axis coordinate) of the long side surfaces 5b-5c of the slab 5.

Here, _ht ( _xB ,y,t) is the heat transfer coefficient at the short side surface 5a of the slab 5, and _ht (x, _yB ,t) is the heat transfer coefficient at the long side surfaces 5b-5c of the slab 5. Furthermore, _T ( _xB ,y,t) is the external atmosphere temperature around the short side surface 5a of the slab 5, and _T (x, _yB ,t) is the external atmosphere temperature around the long side surfaces 5b-5c of the slab 5. As will be described in detail later, in this embodiment, the boundary conditions shown in equations (5) and (6) are set based on the results of heat transfer calculation for the surface of the slab 5 at at least one intermediate calculation position that is set between two heat transfer and solidification calculation positions adjacent in the casting direction (see equations (43) and (44)). This is one of the major differences between the heat transfer solidification model used in this embodiment and the heat transfer solidification models described in JP 2019-48322 A and JP 2019-141893 A.

As described in JP 2019-48322 A and JP 2019-141893 A, symmetry may be utilized to set the so-called quarter cross section of the tracking surface from the corner of the slab 5 to the center of the slab 5 as the area to be calculated.

<Relationship between enthalpy, temperature, and solid fraction>
In alloy steel, when the temperature of the steel falls below the liquidus temperature T _L determined by the component concentration, solidification begins and the solid fraction f _s exceeds 0 (f _s > 0). The temperature of the steel then decreases until solidification is complete and the solid fraction f _s becomes 1 (f _s = 1). Considering that the solid fraction f _s is 0≦f _s ≦1, the relationship between enthalpy H and temperature T is expressed by the following formula (7).

where T ₀ is an arbitrary integration constant and T is the temperature at the calculation point on the tracking surface.
<Heat conduction equation with enthalpy as a variable>
When both sides of equation (7) are partially differentiated with respect to x, the following equation (8) is obtained, so that the heat flux _qx in the x-axis direction is expressed by the following equation (9): Similarly, when both sides of equation (7) are partially differentiated with respect to y, the heat flux _qy in the y-axis direction is expressed by the following equation (10):

By substituting equations (9) and (10) into equation (1), the heat conduction equation can be rewritten at the interior points of the tracking surface as the following equation (11) with enthalpy H as a variable. In this manner, in this embodiment, a two-dimensional heat transfer and solidification calculation is performed assuming that heat transfer occurs in the x-axis and y-axis directions.

Several models have been proposed to express the relationship between temperature T and solid fraction _fs . One of them is a model that interpolates the solid fraction _fs between the liquidus temperature T _L at which solidification starts and the solidus temperature at which solidification is completed, as shown in the following formula (12).

The interpolation function φ(T(x,y,t)) is generally monotonically increasing, and can be expressed as a linear function with respect to temperature T or as a quadratic function with respect to temperature T.

Equations (7), (11), and (12) are closed with respect to the variables H, T, and _fs . Therefore, by simultaneously solving these equations, the enthalpy H(x, y, t), temperature T(x, y, t), and solid fraction _fs (x, y, t) in the tracking surface can be obtained.

In the solid-liquid coexistence region, the temperature is determined by the concentration of solute on the liquid phase side at the interface between the solid and liquid phases, so that the following models expressed as the following formulas (13) and (14) may be used as another model expressing the relationship between the temperature T and the solid fraction _fs from the phase diagram, where μ _max is the maximum number of solute components considered in the heat transfer solidification model.

Equations (7), (11), (13), and (14) are closed with respect to the variables H, T, _fs , and C _Lμ . Therefore, by simultaneously solving these equations, the enthalpy H(x,y,t), temperature T(x,y,t), solid fraction _fs (x,y,t), and solute concentration C _Lμ (x,y,t) on the liquid phase side of the solid-liquid interface can be obtained in the tracking plane.

When simultaneously solving equations (7), (11), and (12) (or equations (13) and (14)), a numerical solution is obtained by discretizing the partial differential equations in space and time. That is, in this embodiment, the differential equation of equation (11) is spatially discretized within the tracking surface and on the boundary line of the tracking surface, and a model is used in which the differential equation is updated at discretized times t = 1, 2, ... from the value at t-1 one time before, incorporating the boundary conditions on the surface of the slab 5. Time t = 0 refers to the start of the calculation, and the initial conditions in the continuous time model shown in equations (1) to (14) are set for each variable.

As described above, the tracking planes are set at regular intervals Δz in the casting direction. The number n indicating the heat transfer and solidification calculation position is determined as follows. That is, the molten steel surface position (position of the molten steel meniscus 4) in the mold 1 is set as 0 (n=0), and the positions arranged at regular intervals of Δz in the withdrawal direction (casting direction) are set as 1, 2, ..., n _max (n=1, 2, ..., n _max ). A plurality of calculation points are set inside and on the boundary line of the tracking planes so that they are arranged in the same manner on each tracking plane. When updating state variables from the tracking plane of the heat transfer and solidification calculation position n-1 on the upstream side to the tracking plane of the heat transfer and solidification calculation position n on the downstream side, the discretization interval Δt of time is expressed by the following equation (15).

where v _c is the casting speed.
A model discretized at the coordinates ( _xi , _yj ) of a calculation point on the tracking surface is expressed as in the following formulas (16) and (17) using enthalpy, temperature, solid fraction, and the enthalpy and solid fraction at calculation points adjacent to the calculation point in question. Here, the calculation points adjacent to the calculation point with coordinates ( _xi , _yj ) are the calculation points with coordinates ( _xi+1 , yj ₎ , coordinates ( _xi , _yj+1 ), coordinates (xi _{- 1} , _yj ), and coordinates (xi, _yj-1 ).

Here, N(i,j) represents a set of calculation points adjacent to the calculation point (coordinates (x _i , y _j )). Also, a ^t _i,j is a coefficient multiplied by H _i,j, t when the partial differential equation of formula (11) is discretely approximated by central difference and rearranged into the form of formula (16). Similarly, α ^t _i,j is a coefficient multiplied by f _si,j,t . The coefficient a ^t _i,j includes the thermal conductivity λ _x or λ _y , specific heat ρ, and solidification latent heat L _h that appear in formula (11). The coefficient α ^t _i,j includes the thermal conductivity λ _x or λ _y , specific heat c, and density ρ that appear in formula (11). Also, b _i,j is always 0 (zero) when the calculation point of coordinates (x _i , y _j ) is not a point on the boundary line of the tracking surface. On the other hand, when the coordinates (x _i , y _j ) are points on the boundary line of the tracking surface, b _i,j is a coefficient multiplied by (T _a -ωT _i,j,t -(1-ω)T _i,j,t-1 ) when the partial differential equation of equation (11) is discretely approximated by central difference and rearranged into the form of equation (16), and includes the heat transfer coefficient h _t (x _B , y, t-1) or h _t (x, y _B , t-1) in equations (5) and (6).

In addition, in equation (16), the calculation stabilization parameter ω is a variable that takes values between 0 and 1 (0≦ω≦1). When updating enthalpy, if ω<1, and the time step Δt is long, the calculation results may become unstable and diverge, and it is known that if ω is small, the calculations tend to become unstable. For this reason, the calculations are stabilized by setting an appropriate ω. ω=0 means that the explicit method is used, ω=1 means that the implicit method is used, and ω=1/2 means that the semi-explicit semi-implicit method (Crank-Nicholson method) is used.

Moreover, p in T _p,t and H _p,t in formula (17) is an abbreviation for the coordinates (x _i , y _j ) of the calculation point. The range of values that the temperature T can take is divided into a plurality of temperature sections in advance. k(p) is the number of a temperature section that includes the temperature T(p) and satisfies T ^k(p) ≦ T(p) < T ^k(p)+1 . Moreover, I _c (T ^k(p) ) is a value that is calculated in advance by performing the integration in formula (7) in the range from temperature T ₀ to the boundary value T ^k(p) . ^{c *} _k(p)+1/2 is a representative value of the specific heat c of molten steel in temperature section k(p), and for example, the average value of the specific heat c of molten steel in temperature section k(p) is used as the representative value c ^* _k(p)+1/2 . It should be noted that c ^* _k(p)+1/2 corresponds to the symbol with an * placed above the k in c ^* _k(p)+1/2 in equation (17).

(Heat transfer coefficient model)
Fig. 2 is a diagram for explaining an example of a cooling method for the slab 5. In Fig. 2, a region of the slab 5 on the side of the support rolls 7l, 7m between two support rolls 7l, 7m adjacent to each other in the casting direction is taken out, and differences in cooling methods for the slab 5 in a cross section (y-z cross section in Fig. 1) parallel to the thickness direction and the casting direction are shown as an example.

In FIG. 2, region a is the region of the long side surface 5b of the slab 5 that comes into contact with the support rolls 7l and 7m. Region a is mainly cooled by contact with the support rolls 7l and 7m. Region b is the region of the long side surface 5b of the slab 5 that does not come into contact with the support rolls 7l and 7m and the cooling water. Region b is mainly cooled by radiation heat dissipation. Region c is the region of the long side surface 5b of the slab 5 that comes into contact with the cooling water sprayed from the cooling spray 2k in the form of a water film, but does not directly collide with the air sprayed from the cooling spray 2k. Region c is mainly cooled by the water film generated by the cooling water. Region d is the region of the long side surface 5b of the slab 5 that comes into direct contact with the cooling water and air sprayed from the cooling spray 2k in the form of mist. Region d is mainly cooled by the cooling water in the form of mist. Region e is a region where cooling water flowing from above as a water film accumulates between the support roll 7m and the slab 5. Region e is mainly cooled by the water that accumulates between the support roll 7m and the slab 5. As described above, there are multiple regions a to e with different cooling methods between two support rolls 7l and 7m that are adjacent in the casting direction. Regions a to e exist in all of the regions between two support rolls that are adjacent in the casting direction among the support rolls 7a to 7v. However, when the slab 5 (casting direction) is horizontal, region e does not exist.

In regions a, c, and e, a constant heat transfer coefficient is often used. Meanwhile, in region b, a heat transfer coefficient _ht expressed by the following formula (18) based on the Stefan-Boltzmann law for radiation heat dissipation is used. In region d, a heat transfer coefficient _ht expressed by the following mathematical model formula (19) obtained based on model experiments is used.

Here, ε is the emissivity, and σ is the Stefan-Boltzmann constant. T is the surface temperature of the slab 5, and T _a is the external atmospheric temperature of the slab 5. A is a coefficient determined according to the surface temperature condition of the slab 5, and W is the flow density of the cooling water impinging on the slab 5. f and g are dimensionless constants. The constant g is a negative value. Therefore, in region d, the lower the surface temperature T of the slab 5, the larger the heat transfer coefficient h _t becomes. In addition, in region d, the relationship between the heat transfer coefficient h _t and the surface temperature T of the slab 5 becomes nonlinear, so that the magnitude of the heat flux q _y from the slab 5 to the outside becomes a nonlinear characteristic with respect to the surface temperature T of the slab 5.

(idea)
Fig. 3 is a diagram showing an example of calculated values of the heat transfer coefficient in each of the regions a to e. When comparing the magnitude of the heat transfer coefficients in the regions a to e, as shown in Fig. 3, the heat transfer coefficients in the regions c to e where the amount of cooling water is large and the heat transfer coefficient of the region a in contact with the support rolls 7l and 7m can be several tens of times larger than the heat transfer coefficient of the region b.

The length in the casting direction of each of the regions a to e is, for example, as follows. That is, the length in the casting direction of the region a is about 10 mm. The length in the casting direction of the region d is about 15 to 30 mm, depending on the shape of the nozzles of the cooling sprays 2a to 2k. When the slab 5 (casting direction) is not horizontal, the length in the casting direction of the region c above the region d is about 10 to 50 mm, and the region c below the region d extends from the lower end of the region d to the upper end of the region e (the region c below the region d is defined as the region sandwiched between the regions d and e). The length in the casting direction of the region e is about 10 to 30 mm. The length in the casting direction of the region b is the length of the remaining part between the two support rolls adjacent in the casting direction. As mentioned above, when the slab 5 (casting direction) is horizontal, the region e does not occur, and in this case, the length in the casting direction of the region c is about 20 to 50 mm.

On the other hand, the casting direction interval Δz between two adjacent heat transfer solidification calculation positions in the casting direction is set to a minimum value of, for example, about 50 mm. The shorter the casting direction interval Δz between the heat transfer solidification calculation positions, the greater the number of tracking surfaces on which the heat transfer solidification calculation is performed simultaneously, and the greater the calculation load. On the other hand, the greater the casting direction interval Δz between the heat transfer solidification calculation positions, the greater the interval between the two adjacent tracking surfaces in the casting direction, and the lower the calculation accuracy. As shown in FIG. 1, when the estimation result by the estimation device 100a is used to control the flow rate of the cooling water by the control device 100b during continuous casting, it is necessary to perform the estimation under time constraints according to the casting speed, and the casting direction interval Δz between the heat transfer solidification calculation positions is determined by balancing the calculation accuracy (estimation accuracy) and the calculation load (calculation time).

When the distance Δz between two adjacent heat transfer solidification calculation positions in the casting direction is 50 mm, there may be an area cooled by multiple cooling methods between two adjacent heat transfer solidification calculation positions in the casting direction (i.e., two tracking surfaces). For this reason, if the difference in the heat transfer coefficient due to the difference in cooling method is not accurately evaluated, there is a risk of a large error in the calculated value of the temperature of the slab 5. Since the difference in cooling method has a particularly large effect on the surface temperature of the slab 5, there is a risk of a particularly large error in the calculated value of the surface temperature of the slab 5. The number of cooling methods between two adjacent heat transfer solidification calculation positions in the casting direction may be one, two, or three or more, depending on the location and distance of the heat transfer solidification calculation positions.

As described above, when there are multiple regions (e.g., region b and region a, region b and region d) between two adjacent heat transfer solidification calculation positions in the casting direction, in which the heat transfer coefficient differs by several tens of times, there is a risk that the calculation accuracy will vary greatly depending on whether or not the difference in heat transfer coefficient due to the difference in cooling method is taken into account. Furthermore, when there is a region between two adjacent heat transfer solidification calculation positions in the casting direction, such as region d, in which the heat transfer coefficient is large and depends on the surface temperature of the slab 5, there is a risk that the heat removal from the slab 5 due to cooling cannot be evaluated with high accuracy unless the change in the heat transfer coefficient due to the change in the surface temperature of the slab 5 is taken into account. To address this issue, it is possible to shorten the interval Δz in the casting direction between the heat transfer solidification calculation positions, but shortening the above interval Δz increases the calculation load, and there is a risk that the heat transfer solidification calculation for each tracking surface will not be completed while the casting progresses by the interval Δz.

Therefore, in this embodiment, in the heat transfer and solidification calculation based on the formula (16) and the formula (17) for one tracking surface moving between two adjacent heat transfer and solidification calculation positions in the casting direction, at least one intermediate calculation position is set between these two adjacent heat transfer and solidification calculation positions. Then, at the intermediate calculation position, the heat transfer calculation for the surface of the slab 5 is performed using the result of the heat transfer and solidification calculation for the tracking surface at the upstream heat transfer and solidification calculation position, and the result of the heat transfer calculation is used to perform the heat transfer and solidification calculation for the tracking surface when the tracking surface at the upstream heat transfer and solidification calculation position moves to the downstream heat transfer and solidification calculation position. In this way, by performing the heat transfer calculation at the intermediate calculation position, it is possible to take into account the temperature change of the surface of the slab 5 and the area nearby between the two adjacent heat transfer and solidification calculation positions in the casting direction. Therefore, when performing the heat transfer and solidification calculation for the tracking surface at the downstream heat transfer and solidification calculation position n, the heat transfer coefficient according to the temperature change can be used, so that the estimation accuracy of the state (temperature and solid fraction) of the slab during casting can be improved. Furthermore, at the intermediate calculation positions, heat transfer calculation is performed on the surface of the slab 5, so there is no need to set a wide area (tracking surface) as the calculation target area as in the heat transfer and solidification calculation at the heat transfer and solidification calculation position, and there is no need to calculate the solid fraction _fs . Therefore, the calculation load at the intermediate calculation positions is lighter than the calculation load for the entire tracking surface at the heat transfer and solidification calculation position, so that it is possible to improve the estimation accuracy of the slab state while suppressing an increase in the calculation load. The estimation device 100a of the present embodiment described below has been made based on this idea.

(Estimation device 100a, control device 100b)
4 is a diagram showing an example of the functional configuration of the estimation device 100a and the control device 100b. The hardware of the estimation device 100a and the control device 100b is realized by using, for example, an information processing device including a processor (e.g., a CPU), a ROM, a RAM, and various interfaces, or dedicated hardware. An example of the functions of the estimation device 100a and the control device 100b of this embodiment will be described below.

First, an example of the functions of the estimation device 100a will be described.
<Operation Data Acquisition Unit 410>
The operation data acquisition unit 410 acquires operation data of the continuous casting machine. The operation data of the continuous casting machine includes, for example, the size of the slab 5 in a direction perpendicular to the casting direction, the casting speed v _c , the temperature of the molten steel in the mold 1, the concentration of solute components in the molten steel, the liquidus temperature T _L of the molten steel calculated using the concentration of the solute components, the amount of cooling water sprayed from the cooling sprays 2a to 2t arranged in each cooling zone of the secondary cooling zone, and the cooling conditions at each point on the surface of the slab 5. The operation data acquisition unit 410 acquires the operation data of the continuous casting machine by, for example, receiving the operation data of the continuous casting machine from a higher-level process computer or a lower-level instrumentation device (not shown).

The cooling conditions include, for example, data indicating the range over which the cooling water sprayed from the cooling sprays 2a to 2t collides with the surface of the slab 5. The operation data acquisition unit 410 calculates the distribution of the flow rate density W of the cooling water over the entire surface of the slab 5 using data indicating the amount of cooling water sprayed from the cooling sprays 2a to 2t, data indicating the range over which the cooling water sprayed from the cooling sprays 2a to 2t collides with the surface of the slab 5, and the number and arrangement of the cooling sprays 2a to 2t.

<Heat transfer solidification calculation position setting unit 420>
The heat transfer solidification calculation position setting unit 420 sets the position z of the slab 5 in the casting direction as the heat transfer solidification calculation position, so that the interval in the casting direction becomes the interval at which the tracking surfaces occur in the casting direction, with the position of the molten steel meniscus 4 in the mold 1 being z=0. In this embodiment, the interval at which the tracking surfaces occur is Δz shown in formula (15), which is a predetermined constant interval.

The heat transfer solidification calculation position setting unit 420 sets the calculation points of the tracking surface at the heat transfer solidification calculation positions. FIG. 5 is a conceptual diagram showing an example of calculation points 511a, 511b, 521a, 521b, 531a to 534a, 531b to 534b of the tracking surface at two heat transfer solidification calculation positions n-1 and n adjacent to each other in the casting direction. FIG. 5 also shows a temperature calculation position k set between the heat transfer solidification calculation positions n-1 and n. As described above, n=0 is given to the heat transfer solidification calculation position that coincides with the molten steel surface position (position of the molten steel meniscus 4) in the mold 1, and n=1, 2, ..., n _max are given to the heat transfer solidification calculation positions arranged at regular intervals Δz along the withdrawal direction (casting direction) in order from the heat transfer solidification calculation position on the upstream side. n=n _max indicates the most downstream heat transfer solidification calculation position (the position of the machine end outlet 9 in this embodiment). In Fig. 5, among these heat transfer solidification calculation positions n = 0, 1, 2, ..., n _max , two arbitrary heat transfer solidification positions n-1, n adjacent in the casting direction are shown. The number of pairs of two heat transfer solidification positions n-1, n adjacent in the casting direction is n _max .

In FIG. 5, there are tracking surfaces 500a and 500b at two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction, and the state immediately after the heat transfer solidification calculation is performed at time t-Δt is shown. In order to perform spatially discretized heat transfer solidification calculation within the tracking surface and on the boundary line of the tracking surface, the heat transfer solidification calculation position setting unit 420 divides each tracking surface 500a and 500b into a lattice shape, and sets calculation points 511a, 511b, 521a, 521b, 531a to 534a, and 531b to 534b one by one in each divided rectangular area. In FIG. 5, calculation points 511a, 511b, 521a, 521b, 531a to 534a, and 531b to 534b are indicated by black circles (●) (the symbols are omitted to avoid complication of notation, but black circles (●) without symbols also indicate calculation points). As shown in FIG. 5, the sizes of the rectangular areas set on the tracking surfaces 500a and 500b do not have to be the same.

The heat transfer solidification calculation position setting unit 420 sets multiple heat transfer solidification calculation positions n in the above manner, and sets multiple rectangular areas and multiple calculation points 511a, 511b, 521a, 521b, 531a to 534a, 531b to 534b at each of the multiple heat transfer solidification calculation positions n.

<Intermediate calculation position setting section 430>
The intermediate calculation position setting unit 430 sets an interval between one or more (plural in FIG. 5) intermediate calculation positions k set between two heat transfer solidification calculation positions n-1 and n adjacent to each other in the casting direction. For example, the intermediate calculation position setting unit 430 may set the interval to an interval designated through an input by a user, or may calculate the interval of the intermediate calculation positions k based on the number of intermediate calculation positions k designated similarly and Δz. In this embodiment, a case where a plurality of intermediate calculation positions k=0 to N are set at equal intervals is shown as an example. FIG. 1 is a diagram showing an example of a relationship with a calculation position k. k is a number indicating an intermediate calculation position set in a section between two heat transfer solidification calculation positions n-1 and n adjacent to each other in the casting direction. As shown in Fig. 5 and Fig. 6, the upstream heat transfer solidification calculation position n-1 is k=0, and k=1, 2, ..., N in the drawing direction (casting direction). The downstream heat transfer and solidification calculation position n is set to k = N.

In Fig. 6, the tracking surfaces (cross sections for heat transfer and solidification calculation) at each heat transfer and solidification calculation position n are shown by solid lines perpendicular to the casting direction. Here, at time t-Δt, the heat transfer and solidification calculation positions n-1 and n shown in Fig. 5 are assumed to be 2 (n=2), and tracking surfaces 500a and 500b shown in Fig. 5 are shown in Fig. 6. There are also tracking surfaces at the other heat transfer and solidification calculation positions n (n=0, 3, ..., n _max -1, n _max ). In Fig. 6, the tracking surfaces at the heat transfer and solidification calculation positions n=0, n _max -1, and n _max at time t-Δt are denoted as tracking surfaces 500c, 500d, and 500e.

The cross section 600 for heat transfer calculation at each intermediate calculation position k is shown by a dashed line perpendicular to the casting direction. The cross section 600 for heat transfer calculation is a cross section of the slab 5 when the slab 5 is cut in a direction perpendicular to the casting direction at each intermediate calculation position k. Note that the cross section 600 for heat transfer calculation at intermediate calculation positions k = 0, N overlaps with each of the tracking surfaces (e.g., tracking surfaces 500a, 500b) at heat transfer solidification calculation positions n-1 and n, and is therefore shown by a solid line perpendicular to the casting direction.

The intermediate calculation position setting unit 430 sets intervals between a plurality of intermediate calculation positions _k in each section having two adjacent heat transfer solidification calculation positions n-1 and n at both ends in the casting direction among the heat transfer solidification calculation positions n=0 to n max.

As shown in FIG. 6, for example, the heat transfer solidification calculation position n=1 is the downstream heat transfer solidification calculation position n relative to the heat transfer solidification calculation position n=0. Therefore, for the tracking surface 500c at the heat transfer solidification calculation position n=0, the tracking surface 500a at the heat transfer solidification calculation position n=1 is the downstream tracking surface. Therefore, in the section with the heat transfer solidification calculation positions n=0 and 1 at both ends (when performing the heat transfer solidification calculation for the tracking surface 500c until the tracking surface 500c moves to the heat transfer solidification calculation position n=1), the heat transfer solidification calculation position n=1 coincides with the intermediate calculation position k=N (see FIG. 5).

On the other hand, the heat transfer solidification calculation position n=1 is the upstream heat transfer solidification calculation position n-1 relative to the heat transfer solidification calculation position n=2. Therefore, relative to the tracking surface 500b at the heat transfer solidification calculation position n=2, the tracking surface 500a at the heat transfer solidification calculation position n=1 is the upstream tracking surface 500. Therefore, in the section with the heat transfer solidification calculation positions n=1 and 2 at both ends (when performing the heat transfer solidification calculation for the tracking surface 500a until the tracking surface 500a moves to the heat transfer solidification calculation position n=2), the heat transfer solidification calculation position n=1 coincides with the intermediate calculation position k=0.

Here, the movement of the tracking surfaces 500a to 500d means that the position of the tracking surfaces 500a to 500d is moved from the heat transfer solidification calculation position n-1 where the tracking surfaces were located to the heat transfer solidification calculation position n, including the result of the heat transfer solidification calculation for the tracking surfaces 500a to 500d. Note that since there is no heat transfer solidification calculation position downstream of the heat transfer solidification calculation position n = n _max , moving the tracking surface 500e at the heat transfer solidification calculation position n = n _max is equivalent to eliminating the tracking surface 500e.

As described above, the heat transfer solidification calculation position n=1 may be the intermediate calculation position k=0 or may be the intermediate calculation position k=N.
Similarly to the heat transfer solidification calculation position n=1, the heat transfer solidification calculation position n=2, . . . , n _max-1 may be the intermediate calculation position k=0 or the intermediate calculation position k=N.

The most upstream heat transfer solidification calculation position n=0 becomes the intermediate calculation position k=0 in the section with heat transfer solidification calculation positions n=0 and 1 at both ends. There is no heat transfer solidification calculation position n upstream of the most upstream heat transfer solidification calculation position n=0. Therefore, the most upstream heat transfer solidification calculation position n=0 will never become the intermediate calculation position k=N.

In addition, the most downstream heat transfer solidification calculation position n= _nmax is the intermediate calculation position k=N in the section with the heat transfer solidification calculation positions n= _nmax-1 and _nmax at both ends. There is no heat transfer solidification calculation position n downstream of the most downstream heat transfer solidification calculation position n= _nmax . Therefore, the most downstream heat transfer solidification calculation position n= _nmax will never be the intermediate calculation position k=0.

As described above, the intermediate calculation position k = 0, N coincides with the heat transfer solidification calculation position n. Therefore, at the same intermediate calculation position k = 0, N (the same heat transfer solidification calculation position n), there may be cases where the same variable (e.g., the surface temperature of the slab) is calculated at the same time by the heat transfer calculation by the heat transfer calculation unit 440 (described later) and the heat transfer solidification calculation by the heat transfer solidification calculation unit 470 (described later). In this embodiment, in such cases, the result of the heat transfer solidification calculation by the heat transfer solidification calculation unit 470 is given priority and adopted.

In the following, a case where heat transfer solidification calculation is performed on the tracking surface 500a will be exemplified. For the other tracking surfaces 500b to 500e, processing is performed in parallel at the same time t as for the tracking surface 500a.
As described with reference to FIG. 2 and FIG. 3, there may be a plurality of regions a to e with different cooling methods between two support rolls 7l and 7m adjacent in the casting direction. The intermediate calculation position k is set with the objective of reflecting the change in the surface temperature of the slab 5 due to the difference in the cooling method in the heat transfer solidification calculation for the tracking surface 500a at the downstream heat transfer solidification calculation position n. Therefore, it is preferable to set the intermediate calculation position k between two heat transfer solidification calculation positions n-1 and n so that the cooling method at at least one of the two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction is different from the cooling method at at least one intermediate calculation position k set between the two heat transfer solidification calculation positions n-1 and n. By setting the intermediate calculation position k in this way, it is possible to include regions with two or more cooling methods in the region with the two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction at both ends, and it is possible to reflect the change in the surface temperature of the slab 5 due to the difference in the cooling method in the heat transfer solidification calculation for the tracking surface 500a at the downstream heat transfer solidification calculation position. In addition, when the cooling methods at two heat transfer solidification calculation positions n-1 and n adjacent to each other in the casting direction are different, and when both of the cooling methods at the two heat transfer solidification calculation positions n-1 and n are different from the cooling method at at least one intermediate calculation position k set between the two heat transfer solidification calculation positions n-1 and n, the region with the two heat transfer solidification calculation positions n-1 and n as both ends includes a region with three or more types of cooling methods.

The intermediate calculation position k is set as another purpose to improve the accuracy of the heat transfer solidification calculation for the region, such as region d, where the heat transfer coefficient depends on the surface temperature of the slab 5 and the magnitude of the heat flux from the slab 5 to the outside has nonlinear characteristics with respect to the surface temperature of the slab 5. Therefore, it is preferable that the casting direction distance between two adjacent intermediate calculation positions k in the casting direction is equal to or less than 1/2 of the value assumed as the minimum value of the casting direction length of region d. By setting the intermediate calculation position k in this manner, multiple intermediate calculation positions k can be set for region d. Therefore, the change in the surface temperature of the slab 5 in region d can be reflected in the heat transfer solidification calculation for the tracking surface 500a at the downstream heat transfer solidification calculation position n.

The intervals between the multiple intermediate calculation positions k = 0 to N do not have to be equal, nor do they have to be fixed values. For example, the interval between the intermediate calculation positions k corresponding to area d may be shorter than the intervals between the other areas a to c and e. Furthermore, the intervals between the multiple intermediate calculation positions k = 0 to N may be changed depending on the content of the operation data.

The intermediate calculation position setting unit 430 sets an intermediate calculation position k between two adjacent heat transfer solidification calculation positions n-1 and n in the casting direction according to the interval set as described above, and sets a calculation point on the heat transfer calculation target cross section 600 at each intermediate calculation position k. The setting of the intermediate calculation position k is performed for each section whose two adjacent heat transfer solidification calculation positions n-1 and n in the casting direction are at both ends.

7 is a conceptual diagram showing an example of calculation points 711, 721, and 731 set for the heat transfer calculation target cross section 600 at the intermediate calculation position k. The intermediate calculation position setting unit 430 divides the heat transfer calculation target cross section 600 into a lattice shape in the same manner as the tracking surface 500a, and sets calculation points 711 and 721 one by one for the rectangular area 710 including the surface of the slab 5 and the rectangular area 720 adjacent (inside) to the rectangular area 710 among the divided rectangular areas (the calculation point 711 is a calculation point set in the rectangular area 710 including the surface of the slab 5, and the calculation point 721 is a calculation point set in the rectangular area 720 adjacent (inside) to the rectangular area 710 including the surface of the slab 5). In this way, the number, size, and arrangement of the rectangular areas 510a and 520a set on the tracking surface 500a are the same as the number, size, and arrangement of the rectangular areas 710 and 720 set on the heat transfer calculation target cross section 600. In the following description, the rectangular region 710 including the surface of the slab 5 is referred to as the rectangular region of the surface layer, and the rectangular region 720 adjacent to the rectangular region 710 including the surface of the slab 5 on the inside is referred to as the rectangular region of the second layer.

In FIG. 7, the calculation points 711 set in the rectangular area 710 on the surface layer of the cross section 600 for heat transfer calculation are arranged in the same manner as the calculation points 511a set in the rectangular area 510a on the surface layer of the tracking surface 500a shown in FIG. 5. Similarly, in FIG. 7, the calculation points 721 set in the rectangular area 720 on the second layer of the cross section 600 for heat transfer calculation are arranged in the same manner as the calculation points 521a set in the rectangular area 520a on the second layer of the tracking surface 500a shown in FIG. 5.

In FIG. 7, the calculation points 711 set in the rectangular area 710 of the surface layer of the cross section 600 for heat transfer calculation are shown by black circles (●), and the calculation points 721 set in the rectangular area 720 of the second layer of the cross section 600 for heat transfer calculation are shown by white circles (◯) (the symbols are omitted to avoid complication of notation, but the black circles (●) and white circles (◯) without symbols also indicate calculation points). The cross section 600 for heat transfer calculation shown in FIG. 7 is set at each of the intermediate calculation positions k = 0 to N shown in FIG. 5. In this embodiment, among the rectangular areas set in the cross section 600 for heat transfer calculation, no calculation points are set in the rectangular area inside the rectangular area 720 of the second layer. In other words, the area inside the rectangular area 720 of the second layer is not the area targeted for heat transfer calculation.

In this embodiment, the intermediate calculation position setting unit 430 sets the external rectangular area 730 as shown in FIG. 7 so that the heat transfer calculation unit 440, which will be described later, can calculate the external atmosphere temperature around the heat transfer calculation target cross section 600 set at the intermediate calculation position k. The external rectangular area 730 is a rectangular area that defines the area around the heat transfer calculation target cross section 600 using virtual lines that extend the sides of the rectangular area set in the heat transfer calculation target cross section 600 outward. The length in the y-axis direction of the rectangular area aligned in the direction (x-axis direction) along the long side surfaces 5b to 5c of the slab 5 out of the external rectangular area 730 is, for example, the length in the y-axis direction of the rectangular area 720 of the second layer. The length in the x-axis direction of the rectangular area aligned in the direction (y-axis direction) along the short side surface 5a of the slab 5 out of the external rectangular area 730 is, for example, the length in the x-axis direction of the rectangular area 720 of the second layer.

The intermediate calculation position setting unit 430 also sets one calculation point 731 for each of the external rectangular areas 730. In FIG. 7, the calculation points for the external rectangular areas 730 are indicated by black circles (●) (the symbols are omitted to avoid complicating the notation, but black circles (●) without symbols also indicate calculation points).

<Heat transfer calculation unit 440>
The heat transfer calculation unit 440 performs heat transfer calculations at time t for the surface of the slab 5 at each intermediate calculation position k.
<<Heat transfer model>>
An example of a heat transfer model that is a calculation model used for the heat transfer calculation executed by the heat transfer calculation unit 440 will be described.

In this embodiment, spatial discretization is performed assuming that the temperature is constant in a range of a length Δy _s in the thickness direction (y-axis direction) of a rectangular region 710 of the surface layer aligned in a direction (x-axis direction) along the long side surfaces 5b to 5c of the slab 5, and that the temperature is constant in a range of a length Δx _s in the width direction (x-axis direction) of the rectangular region 710 of the surface layer aligned in a direction (y-axis direction) along the short side surface 5a of the slab 5. In addition, in Figs. 5 and 7, the temperature of a calculation point 721 set in a rectangular region 720 of the second layer of the heat transfer calculation target cross section 600 at intermediate calculation positions k = 1 to N-1 set between heat transfer and solidification calculation positions n-1 and n is constant at a temperature T _s (x, y, t - Δt) calculated by the heat transfer and solidification calculation unit 470 described later for the calculation point 721 set in a rectangular region 520a of the second layer of the tracking surface 500a. Hereinafter, T _s (x, y, t) will be simply denoted as T _s . Moreover, the temperature _Ts of the calculation point 721 set in the rectangular region 720 of the second layer is referred to as the temperature just below the surface.

Also, the external atmosphere temperature T _{a_k} (x, y, t) around the heat transfer calculation target cross section 600 at the intermediate calculation position k is constant at the external atmosphere temperature T _{a_k-1} (x, y, t) around the heat transfer calculation target cross section 600 at the intermediate calculation position k-1. The external atmosphere temperature T _{a_k-1} (x, y, t) is the external atmosphere temperature calculated by the heat transfer calculation unit 440 (first calculation unit 441) as described below. In the following description, the external atmosphere temperature when the intermediate calculation position k is not specified is simply represented as T _a .

In the following description, for variables other than the external ambient temperature T _a , k, k-1, k-2, etc. will be placed next to the symbol " _{_} " to indicate that the variable is a variable at the intermediate calculation position k, k-1, k-2, etc. Furthermore, for variables other than the external ambient temperature T _a , when the intermediate calculation position k is not specified, the notation of the symbol indicating the intermediate calculation position, such as k, k-1, k-2, etc. placed next to the symbol " _{_} " will be omitted.

In this embodiment, the heat transfer calculation unit 440 calculates the external atmosphere temperature T _a for each external rectangular region 730. The temperature of the calculation point 731 in the external rectangular region 730 becomes the external atmosphere temperature T _a in the external rectangular region 730. However, it is not necessary to calculate the external atmosphere temperature T _a for each external rectangular region 730. For example, among the regions surrounding one heat transfer calculation target cross section 600, a common temperature may be adopted as the external atmosphere temperature T _a for the region surrounding the long side surfaces 5b to 5c of the slab 5, and a common temperature may be adopted as the external atmosphere temperature T _a for the region surrounding the short side surface 5a of the slab 5.

In addition, the density ρ(x, y, t) and specific heat c(x, y, t) of steel are assumed to be constant between two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction. Hereinafter, ρ(x, y, t) and c(x, y, t) will be simply written as ρ and c. In addition, the heat transfer coefficient _ht (x, y, t) will be simply written as _ht .

In the following, in order to avoid the notation of the formulas becoming complicated, the heat transfer coefficient _ht at each calculation point 711, 721 of the heat transfer calculation target cross section 600 is expressed as the discretized heat transfer coefficient _Ht shown in the following formulas (20a) to (20b). Also, the thermal conductivities _λx and _λy in the x-axis direction and y-axis direction at each calculation point 711, 721 of the heat transfer calculation target cross section 600 are expressed as the discretized heat conductivities L shown in the following formulas (21a) to (21b). Note that the discretized heat transfer coefficient _Ht and the discretized heat conductivities L are functions of x, y, and t, but the fact that they are functions of x, y, and t is omitted here.

Here, the formula (20a) and the formula (21a) are used when performing heat transfer calculations for the long side surfaces 5b to 5c of the slab 5. The formula (20b) and the formula (21b) are used when performing heat transfer calculations for the short side surface 5a of the slab 5. Also, as shown in FIG. 7, Δx _c is the length in the x-axis direction from the boundary line of the heat transfer calculation target cross section 600 in the direction parallel to the short side surface 5a of the slab 5 (y-axis direction) to the calculation point 721 immediately below the boundary line among the calculation points 721 of the rectangular region 720 of the second layer. Δy _c is the length in the y-axis direction from the boundary line of the heat transfer calculation target cross section 600 in the direction parallel to the long side surfaces 5b to 5c of the slab 5 (x-axis direction) to the calculation point 721 immediately below the boundary line among the calculation points 721 of the rectangular region 720 of the second layer.

If heat transfer in a direction parallel to the surface of the slab 5 is ignored and one-dimensional heat transfer occurs in a direction perpendicular to the surface of the slab 5, the model equation for the spatially discretized surface temperature T of the slab 5 is expressed by the following equation (22) based on the unsteady one-dimensional heat conduction equation.

When the external atmosphere temperature _T , the subsurface temperature _T , the heat transfer coefficient _h , the density ρ, and the specific heat c are constant in the range from time t to time t+Δt, and 0≦τ≦Δt, the intermediate surface temperature T(t+τ) at time t+τ is expressed by an analytical solution as shown in the following equation (23): Here, the intermediate surface temperature is the surface temperature of the slab 5 at the intermediate calculation position k.

In the second term on the right side of equation (23), ( _HtTa + _LTs )/( _Ht +L) is _the temperature indicating the point where the line segment connecting the point indicating the external ambient temperature _Ta and the point indicating the subsurface temperature _Ts on the coordinate axis indicating temperature is divided internally by the discretized heat transfer coefficient _Ht : discretized thermal conductivity L, and in the following explanation, this is referred to as the intermediate external temperature. The symbol representing the intermediate external temperature is _Tb . Note that since the discretized heat transfer coefficient _Ht and the discretized thermal conductivity L are functions of x, y, and t, the intermediate external temperature _Tb is also a function of x, y, and t.

In this embodiment, the heat transfer calculation unit 440 performs heat transfer calculation based on the formula (23). In the formula (23), one-dimensional heat transfer occurs in a direction perpendicular to the surface of the slab 5, and the intermediate surface temperature T(t+τ) is determined by the external atmosphere temperature T _a and the temperature just below the surface T _s . Therefore, the heat transfer calculation is performed based on the formula (23) assuming that one-dimensional heat transfer occurs in a direction perpendicular to the surface of the slab 5 within the rectangular region 710 of the surface layer, the rectangular region 720 of the second layer, and the outer rectangular region 730. In this embodiment, the heat transfer calculation unit 440 includes a first calculation unit 441, a second calculation unit 442, and a third calculation unit 443. The heat transfer calculation unit 440 increments (increases by 1 in ascending order) the value of k, which indicates the intermediate calculation position to be subjected to the heat transfer calculation, with k=1 as the initial value and k=N as the final value, and repeatedly executes the following processes of the first calculation unit 441, the second calculation unit 442, and the third calculation unit 443 at each time t with a time interval of discretization increment Δt of time. Thus, in this embodiment, the heat transfer calculation unit 440 executes the heat transfer calculation for the surface of the slab 5 at the intermediate calculation position k at time t, with k=0 as the upstream heat transfer and solidification calculation position n-1, using the result of the heat transfer calculation for the surface of the slab 5 at time t-Δt at the upstream heat transfer and solidification calculation position n-1 and the result of the heat transfer calculation for the surface of the slab 5 corresponding to time t-(1-k/N)Δt for the intermediate calculation position downstream of the upstream heat transfer and solidification calculation position n-1 and upstream of the intermediate calculation position k specified by k=1 to N-1.

<<First Calculation Unit 441>>
The first calculation unit 441 identifies to which of the regions a to e the intermediate calculation position k-1, which is set between the heat transfer solidification calculation positions n-1 and n, belongs, based on the distribution of the flow rate density W of the cooling water over the entire surface of the slab 5. Then, the first calculation unit 441 calculates the heat transfer coefficient h _{t_k-1} and the external atmosphere temperature T _{a_k-1} at the intermediate calculation position k-1 according to the identified region. As described above, in this embodiment, the distribution of the flow rate density W of the cooling water over the entire surface of the slab 5 is calculated by the operation data acquisition unit 410.

First, an example of a method for calculating the heat transfer coefficient h _{t_k-1} at the intermediate calculation position k-1 will be described.
The first calculation unit 441 sets a constant heat transfer coefficient h _{t_k-1} set in advance for the intermediate calculation position k-1 belonging to the region a, region c, and region e. The first calculation unit 441 also calculates the heat transfer coefficient h _{t_k-1} for the intermediate calculation position k-1 belonging to the regions b and d using the formulas (18) and (19), respectively. At this time, the intermediate surface temperature T^ _k calculated by the second calculation unit 442 for the intermediate calculation position k-1 (described later) is substituted for T in the formulas (18) and (19) (see formula (24)). Note that T^ _k corresponds to the symbol with ^ added above T in T _k in formula (24). This notation is also the same for the notation of ^ for variables other than T^ _k .

Next, an example of a method for calculating the external ambient temperature T _{a — k-1} at the intermediate calculation position k-1 will be described.
The first calculation unit 441 refers to a table in which the temperature to be adopted as the external ambient temperature T _a is registered in advance for each cooling method (area a to e). An example of the contents registered in this table will be described.

As the external atmosphere temperature T _a in the region a, for example, it is registered that the surface temperature of the support roll in contact with the slab at the intermediate calculation position k-1 is used.
For example, the outside air temperature is registered as the external ambient temperature T _a in the area b.

As the external ambient temperature T _a in the regions c and d, for example, it is registered that the temperature of the cooling water in the cooling spray that is located closest to the intermediate calculation position k-1 among the cooling sprays 2a to 2t is used.

As the external ambient temperature T _a in the region e, for example, it is registered that the temperature of the cooling water in the cooling spray that is located upstream of the support roll 7 m among the cooling sprays 2 a to 2 t and is closest to the intermediate calculation position k-1 is used.
In addition, when a regression equation is used to calculate the external atmosphere temperature T _a as in the example shown here, the regression coefficients may be determined using, for example, the results of a model experiment simulating the secondary cooling zone or the results of a numerical simulation simulating the secondary cooling zone.

In this embodiment, the support rolls 7a to 7v do not contact the narrow side surface 5a of the slab 5. Furthermore, the cooling sprays 2a to 2t are located opposite the long side surfaces 5b to 5c of the slab 5, but are not located opposite the narrow side surface 5a of the slab 5. Therefore, the external atmosphere temperature _{T a} relative to the narrow side surface 5a of the slab 5 may all be the outside air temperature. In other words, the above-mentioned table may be used to calculate the external atmosphere temperature T _a relative to the long side surfaces 5b, 5c of the slab 5.

The first calculation unit 441 calculates the external atmosphere temperature T _{a_k-1} according to the contents registered in the table as described above. When the table is registered as described above, information required to execute the calculation of the contents registered in the table (such as the temperatures of the support rolls 7a to 7v and the outside air temperature) is also acquired as operation data by the operation data acquisition unit 410.

<<Second Calculation Unit 442>>
The intermediate surface temperature T^ _k at the intermediate calculation position k and the intermediate external temperature T _b ^ _{_k-1} at the intermediate calculation position k-1 are expressed as in the following equation (24) based on equation (23).

As shown in equation (24), the intermediate surface temperature T^ _k at intermediate calculation position k is expressed as the temperature indicating the point which divides the line segment connecting the point indicating the intermediate surface temperature T^ _k-1 at intermediate calculation position k-1 and the point indicating the intermediate external temperature _Tb ^ _{_k-1} at intermediate calculation position k-1 internally at γk _-1 :1-γk _-1 on the coordinate axis indicating temperature. Here, γk _-1 and _Tb ^ _{_k-1} are expressed by the following equations (25a) to (29).

Here, formula (25a) and formula (26a) are used when performing heat transfer calculations for the long side surfaces 5b to 5c of the slab 5. Formula (25b) and formula (26b) are used when performing heat transfer calculations for the short side surface 5a of the slab 5. In formula (27), T _{s_n-1} is the temperature just below the surface at the heat transfer and solidification calculation position n-1 on the upstream side (the temperature of the calculation point 721 set in the rectangular region 720 of the second layer), which has already been calculated at time t-Δt by the heat transfer and solidification calculation unit 470 described later. For variables other than the just below the surface temperature T _s, n, N-1, etc. are added next to the symbol " _{_} " to indicate that the variable is a variable at the heat transfer and solidification calculation position n, N-1, etc.

The second calculation unit 442 calculates the intermediate surface temperature T^ _k at the intermediate calculation position k and the intermediate external temperature T _b ^ _k-1 at the intermediate calculation position k-1 by performing the calculations of the formulas (24) to (29). As described above, the intermediate surface temperature T^ _k at the intermediate calculation position k calculated by the second calculation unit 442 is substituted for T in the formulas (18) and (19) in the next calculation of the first calculation unit 441 in which the value of k is incremented and executed. Note that if no region in which the heat transfer coefficient _ht depends on the surface temperature T of the slab 5, such as regions b and d, is included between the heat transfer solidification calculation positions n-1 and n, the calculation of the intermediate surface temperature T^ _k does not need to be executed.

<<Third Calculation Unit 443>>
If heat transfer in a direction parallel to the surface of the slab 5 is ignored and one-dimensional heat transfer occurs in a direction perpendicular to the surface of the slab 5, the following equation (30) is obtained as a discretized heat conduction equation from the upstream heat transfer and solidification calculation position n-1 to the intermediate calculation position k.

Here, T ₀ is the surface temperature of the slab 5 on the tracking surface 500a when the tracking surface 500a is located at the downstream heat transfer solidification calculation position n. β _k-1 is the intermediate temperature change rate parameter at the intermediate calculation position k-1. T _b ^* _{_k-1} is the intermediate average external temperature at the intermediate calculation position k-1. Note that T _b ^* _{_k-1} corresponds to the symbol with an * above the b in T _{b_k-1} in formula (30). This notation is also the same for the * notation of variables other than T _b ^* _{_k-1} .

The intermediate average external temperature T _b ^* _{_k-1} at the intermediate calculation position k-1 represents the average value of the intermediate external temperatures T _b ^{^} _{_0} to T _b ^{^} _{_k} between the upstream heat transfer solidification calculation position n-1 and the intermediate calculation position k. The intermediate temperature change rate parameter β _k-1 at the intermediate calculation position k-1 is a parameter determined based on the discretized heat transfer coefficient H _{t_k-1} and the discretized thermal conductivity L at the intermediate calculation position k-1, and represents the average value of the ease of heat and temperature transfer between the slab 5 and the outside between the upstream heat transfer solidification calculation position n-1 and the intermediate calculation position k.

The surface temperature T ₀ of the slab 5 on the tracking surface 500a when the tracking surface 500a is located at the downstream heat transfer solidification calculation position n is the temperature of each calculation point 511b set in the rectangular region 510b of the surface layer of the tracking surface 500a at the downstream heat transfer solidification calculation position n, and is a function of x, y, and t, but the description of the function of x, y, and t is omitted here. In addition, since the intermediate external temperature T _b is a function of x, y, and t, the intermediate average external temperature T _b ^* is also a function of x, y, and t, but the description of the function of x, y, and t is omitted here. In addition, since the discretized heat transfer coefficient H _t and the discretized thermal conductivity L are functions of x, y, and t, the intermediate temperature change rate parameter β is also a function of x, y, and t, but the description of the function of x, y, and t is omitted here.

For the k-1st intermediate calculation position k-1 counting from the upstream side, equation (30) is expressed as the following equation (31).

In the case of k≧2, the following equation (32) is obtained by comparing ^the coefficients applied to T ₀ on the right-hand side of the following equation obtained by eliminating T ^{^} _{k -1 from} equations (24) and (31), setting the left side to T ^{^} _k -T ₀ , and rearranging the right side as a linear expression of T ₀ , with the left side of equation (30) set to T ^ k -T 0, and rearranging the right side as a linear expression of T _0. Similarly, the following equation (33) is obtained by comparing the constant terms on the right-hand sides of these equations.

On the other hand, when k = 1, the discretized heat conduction equation from the upstream heat transfer and solidification calculation position n-1 to the intermediate calculation position k coincides with equation (24), so the following equations (34) and (35) are obtained.

The third calculation unit 443 calculates the intermediate temperature change rate parameter β _k-1 at the intermediate calculation position k-1 by performing the calculation of equation (32) or equation (34), and calculates the intermediate average external temperature T _b ^* _{_k-1} at the intermediate calculation position k-1 by performing the calculation of equation (33) or equation (35).
As described above, the processes of the first calculation unit 441, the second calculation unit 442, and the third calculation unit 443 are executed for each value of k, with k=1 being the initial value and k=N being the final value.

<Average heat transfer coefficient calculation unit 450>
The average heat transfer coefficient calculation unit 450 calculates the average heat transfer coefficient h _{t_ave} at each time t of the time interval of the discretization increment Δt using the result of the heat transfer calculation in the heat transfer calculation unit 440. The average heat transfer coefficient h _{t_ave} is an average value of the heat transfer coefficient between the slab 5 and the outside between the heat transfer and solidification calculation position n-1 on the upstream side and the heat transfer and solidification calculation position n on the downstream side. In this embodiment, the average heat transfer coefficient calculation unit 450 calculates the average heat transfer coefficient h _{t_ave} by performing calculations of the following formulas (36a) and (36b) using the intermediate temperature change rate parameter β _N-1 finally calculated in the heat transfer calculation unit 440.

Equation (36a) is used when calculating the average heat transfer coefficient _{ht_ave} for the long side surfaces 5b to 5c of the slab 5. Equation (36b) is used when calculating the average heat transfer coefficient _{ht_ave} in the direction parallel to the short side surface 5a of the slab 5 (y-axis direction).

Equations (36a) and (36b) are derived as follows.
If the discretized heat transfer coefficient obtained by substituting the average heat transfer coefficient _{ht_ave} for _ht in equation (20a) or equation (20b) is denoted as _{Ht_ave} , the intermediate surface temperature _TN can be expressed as the following equation (37) from equation (23).

Furthermore, if k = N in equation (30), the following equation (38) is obtained.

When the right-hand side of equation (37) is transformed so that the left-hand side is T^ _N - _T0 and the right-hand side is multiplied by a coefficient of ( _T0 - _Tb ^* _{_N} ) and compared with the right-hand side of equation (38), the following equation (39) is obtained. Then, by transforming equation (39), the following equation (40) is obtained. By applying the right-hand side of equation (40) to the left-hand sides of equations (20a) and (20b), equations (36a) and (36b) are obtained.

<Average External Ambient Temperature Calculation Unit 460>
The average external atmosphere temperature calculation unit 460 calculates the average external atmosphere temperature T _{a_ave} at each time t of the time interval of the discretization step Δt, using the result of the heat transfer calculation in the heat transfer calculation unit 440 and the average heat transfer coefficient h _{t_ave} . The average external atmosphere temperature T _{a_ave} is an average value of the external atmosphere temperature T _a of the slab 5 between the heat transfer solidification calculation position n-1 on the upstream side and the heat transfer solidification calculation position n on the downstream side. In this embodiment, the average external atmosphere temperature calculation unit 460 calculates the average external atmosphere temperature T a_ave by performing calculations of the following formulas (41a) and (41b) using the intermediate average external temperature _{T b} ^* _{_N-1} finally calculated in the heat transfer calculation unit 440 and the average heat transfer coefficient h _{t_ave} .

The formula (41a) is used when calculating the average external atmosphere temperature T _{a_ave} for the long side surfaces 5b to 5c of the slab 5. The formula (41b) is used when calculating the average external atmosphere temperature T _{a_ave} for the narrow side surface 5a of the slab 5.

Equation (41a) and equation (41b) can be obtained by substituting the right-hand sides of equation (20a) and equation (20b) for H _{t_ave} , which is obtained by transforming the following equation (42) obtained based on equation (27) so that the left-hand side becomes T _{a_ave} (where h _t on the right-hand sides of equations (20a) to (20b) is set to h _{t_ave} ).

<Heat Transfer Solidification Calculation Unit 470>
The heat transfer solidification calculation unit 470 executes heat transfer solidification calculation for the tracking surface 500a located at the upstream heat transfer solidification calculation position n-1 at time t-Δt using the result of the heat transfer calculation in the heat transfer calculation unit 440 at each time t in the time interval of the discretized time increment Δt. In this embodiment, the heat transfer solidification calculation unit 470 calculates heat fluxes q _x (x _B , y; t) and q _y (x, y _B ; t) on the surface of the slab 5 in the heat transfer solidification calculation until the tracking surface 500a reaches the heat transfer solidification calculation position n, based on the average heat transfer coefficients h _{t_ave} (x _B , y; t) and h _{t_ave} (x, y _B ; t) and the average external atmospheric temperatures T _{a_ave} (x _B , y; t) and _T a_ave (x, y _B ; t). The semicolon ";" in the parentheses in the notation of the heat transfer coefficient, the external average temperature, and the heat flux here means that it is an average from time t-Δt to t. The heat transfer solidification calculation unit 470 uses the temperature T(x, y, t-Δt) and the solid fraction _fs (x, y, t-Δt) at each calculation point 511a, 521a, 531a to 534a at the heat transfer solidification calculation position n-1 as initial conditions, and uses the heat fluxes _qx ( _xB , y; t) and _qy (x, _yB ; t) on the surface of the slab 5 as boundary conditions, and calculates the temperature T(x, y, t) and the solid fraction _fs (x, y, t) at each calculation point 511a, 521a, 531a to 534a at the time when the tracking surface 500a reaches the downstream heat transfer solidification calculation position by the above-mentioned heat transfer solidification model. Therefore, in this embodiment, for equations (5) and (6), the following equations (43) and (44) are used as boundary conditions in which _ht ( _xB ,y,t) is changed to _{ht_ave} ( _xB , _y ;t), _ht (x, _yB ,t) is changed to _{ht_ave} (x, _yB ;t), T( _xB , _y ,t) is changed to _T0 (xB,y,t-Δt), _T (x, _yB ,t) is changed to _T0 ( _xB ,y,t-Δt), _Ta (xB,y,t) is changed to _{Ta_ave} ( _xB ,y;t), and Ta(x, _yB ,t) is changed to _{Ta_ave} (x, _yB ;t).

In this embodiment, when the control device 100b compares the target value and the calculated value of the temperature at a predetermined position of the slab 5 as described later, the calculation points 511a, 511b, 521a, 521b, 531a to 534a, and 531b to 534b at the predetermined position or the closest position to the predetermined position are included in the calculation points to be calculated for the temperature T(x,y,t) and the solid fraction _fs (x,y,t). For example, it is preferable to include at least one of the calculation points 531a to 534a at the center position or the closest position to the center of the tracking surface 500a and the calculation point 511a at the surface position or the closest position to the surface in the calculation points to be calculated for the temperature T(x,y,t) and the solid fraction _fs (x,y,t).

When the heat transfer solidification calculation unit 470 performs a heat transfer solidification calculation for the tracking surface 500a that was at the upstream heat transfer solidification calculation position n-1 at time t-Δt and obtains the result when it reaches the downstream heat transfer solidification calculation position n at time t, it uses the result of the heat transfer solidification calculation for the tracking surface 500a (the result of the heat transfer solidification calculation for the tracking surface 500a at time t) as the initial value of the heat transfer solidification calculation for the tracking surface 500a at the heat transfer solidification calculation position n+1 at the next time, time t+Δt. In other words, the heat transfer solidification calculation unit 470 uses the result of the heat transfer solidification calculation for the tracking surface 500a at time t as the initial condition for the heat transfer solidification calculation for the tracking surface 500a that starts from the heat transfer solidification calculation position n at time t+Δt.

In this way, the result of the heat transfer solidification calculation of the tracking surface 500a located at the upstream heat transfer solidification calculation position n-1 at time t is used as the initial value of the heat transfer solidification calculation of the same tracking surface 500a starting from the downstream heat transfer solidification calculation position n at time t+Δt. The moving distance at this time is a distance Δz based on the discretization interval Δt of time and the casting speed v _c (see formula (15)). In this embodiment, the interval in the casting direction between the heat transfer solidification calculation positions n-1 and n and the interval in the casting direction generating the tracking surface 500a are both constant intervals Δz. Therefore, at the same time, all tracking surfaces are located at either the heat transfer solidification calculation positions n-1 or n.

Then, at the next time t+Δt, the heat transfer solidification calculation unit 470 generates a new tracking surface as an upstream tracking surface at the heat transfer solidification calculation position n=0, which is the position (z=0) of the molten steel meniscus 4 in the mold 1. For example, in FIG. 6, the result of the heat transfer solidification calculation performed on the tracking surface 500a at the heat transfer solidification calculation position n=1 at time t becomes the initial condition for the heat transfer solidification calculation for the same tracking surface 500a starting from the heat transfer solidification calculation position n=1 at time t+Δt.

In this embodiment, the solid fraction fs(x, y, t) and temperature T(x, y, t ₎ at each calculation point 511a, 521a, 531a to 534a on the tracking surface 500a at the heat transfer solidification calculation position n=0, which is set at the position (z=0) of the molten steel meniscus 4 in the mold 1, are set such that the solid fraction fs is set to 0 at all (x, y), and the temperature T is set to an initial value estimated from the actual measured temperature of the molten steel in the tundish at all (x, y).
The heat transfer solidification calculation unit 470 executes the above-mentioned process by successively updating the time t.

<Output Unit 480>
The output unit 480 outputs information indicating the result of the heat transfer coagulation calculation for the downstream heat transfer coagulation calculation position n executed by the heat transfer coagulation calculation unit 470. In this embodiment, the output unit 480 transmits information including the temperature T(x, y, t) at the calculation point 511a at a predetermined position where the target value of the temperature is set or the calculation points 531a to 534a closest to the predetermined position, among the temperatures T(x, y, t) at the calculation points 511a, 521a, 531a to 534a on the tracking surface 500a after moving to the downstream heat transfer coagulation calculation position n, calculated by the heat transfer coagulation calculation unit 470, to the control device 100b. Note that the communication between the estimation device 100a and the control device 100b may be wired communication or wireless communication.

Next, an example of the functions of the control device 100b will be described.
<Control Unit 490>
The control unit 490 controls the amount of cooling water in the secondary cooling zone of the continuous casting machine so that the deviation between the target value of the temperature at a predetermined position of the slab 5 being cast by the continuous casting machine and the calculated value approaches zero based on the heat transfer and solidification calculation at the downstream heat transfer and solidification calculation position n executed by the estimation device 100a (heat transfer and solidification calculation unit 470). The target value of the temperature at the predetermined position of the slab 5 is preset in the control device 100b based on, for example, the specifications of the slab 5 (for example, the quality of the components contained in the slab 5).

The control unit 490 determines the amount of cooling water for each cooling zone as an operation amount so as to bring the deviation between the target value and the calculated value of the temperature at a predetermined position of the slab 5 at each heat transfer solidification calculation position n closer to zero. When the predetermined position of the slab 5 coincides with the position of the calculation point 511b, the temperature T (x, y, t) of the calculation point 511a on the tracking surface 500a after moving to the downstream heat transfer solidification calculation position n, calculated by the heat transfer solidification calculation unit 470, becomes the calculated value. When the predetermined position of the slab 5 does not coincide with the positions of the calculation points 531a to 534a, the control unit 490 uses the temperature T (x, y, t) of the calculation points 531a to 534a on the tracking surface 500a after moving to the downstream heat transfer solidification calculation position n, calculated by the heat transfer solidification calculation unit 470, to calculate the temperature at the predetermined position and use it as the calculated value. For example, a representative value (e.g., average value) of the temperatures T(x, y, t) of calculation points 531ba-534a on tracking surface 500a after moving to downstream heat transfer solidification calculation position n is used as the calculation value. Then, control unit 490 sets and outputs water volume instruction values for each of cooling sprays 2a-2t based on the calculated cooling water volume. Flow rate adjustment valves 3a-3e operate so that the amount of cooling water sprayed from cooling sprays 2a-2t is a flow rate corresponding to the water volume instruction value.

The amount of cooling water for each cooling zone is controlled by executing feedback control such as PI control. The specified position of the slab 5 may be any position set between the heat transfer solidification calculation positions, and may be, for example, the surface or center of the slab 5.

(flowchart)
Next, an example of a continuous casting estimation method using the continuous casting control device 100 will be described with reference to the flowchart of Fig. 8. In this embodiment, the continuous casting estimation method using the continuous casting control device 100 is realized by using an estimation method using the estimation device 100a and a control method using the control device 100b.
First, in step S801, the estimating device 100a sets time t to an initial time t _i (e.g., 0 or the current time). The initial time t _i is the time at which processing in the estimating device 100a starts, and means, for example, the timing at which an operator executes an operation to instruct the estimating device 100a to start processing. The following steps S802 to S815 are executed as processing for time t.
Next, in step S802, the operation data acquisition unit 410 acquires operation data of the continuous casting machine.

Next, in step S803, the heat transfer solidification calculation position setting unit 420 sets the heat transfer solidification calculation position n and the calculation points 511a, 511b, 521a, 521b, 531a to 534a, 531b to 534b of each heat transfer solidification calculation position n. Note that if the heat transfer solidification calculation position n and the calculation points 511a, 511b, 521a, 521b, 531a to 534a, 531b to 534b of the heat transfer solidification calculation position n are constant regardless of time t, the process of step S803 may be executed before step S801.

Next, in step S804, the intermediate calculation position setting unit 430 sets an interval of a plurality of intermediate calculation positions k between two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction. Then, the intermediate calculation position setting unit 430 sets the intermediate calculation positions k according to the set interval k, and sets calculation points 711, 721, and 731 on the heat transfer calculation target cross section 600 at each intermediate calculation position k. The process of step S804 is executed for each of two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction among the heat transfer solidification calculation positions n=0 to n _max .

Next, in step S805, the heat transfer calculation unit 440 sets k to 1.
Next, in step S806, the first calculation unit 441 calculates the heat transfer coefficient h _{t_k-1} and the external ambient temperature T _{a_k-1} at the intermediate calculation position k-1.

Next, in step S807, the second calculation unit 442 executes the calculations of equations (24) to (29) to calculate the intermediate surface temperature T^ _k at the intermediate calculation position k and the intermediate external temperature T _b ^ _{_k-1} at the intermediate calculation position k-1.
Next, in step S808, the third calculation unit 443 calculates the intermediate temperature change rate parameter β _k-1 at the intermediate calculation position k-1 by performing the calculation of equation (32) or equation (34). Also, the third calculation unit 443 calculates the intermediate average external temperature T _b ^* _{_k-1} at the intermediate calculation position k-1 by performing the calculation of equation (33) or equation (35).

Next, in step S809, the heat transfer calculation unit 440 determines whether the value of k is N. N is a value obtained by adding 1 to the number of intermediate calculation positions k between two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction. If the result of this determination is that the value of k is not N (NO in step S809), the process of step S810 is executed. In step S810, the heat transfer calculation unit 440 increments the value of k. Then, the processes of steps S806 to S809 are executed for the incremented k.

The processing of steps S805 to S810 is performed for each of the two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction among the heat transfer solidification calculation positions n=0 to n _max . If the number of intermediate calculation positions k differs between the two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction, for example, the value obtained by adding 1 to the number of intermediate calculation positions k at the heat transfer solidification calculation positions n-1 and n having the largest number of intermediate calculation positions k set between them among the two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction is set as N. In this case, the heat transfer calculation unit 440 ends the processing of steps S806 to S810 for the heat transfer solidification calculation positions n-1 and n for which the processing of steps S806 to S810 has been completed for all k among the two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction. Then, the heat transfer calculation unit 440 continues the process of steps S806 to S810 only for the heat transfer solidification calculation positions n-1 and n, which are adjacent to each other in the casting direction, and for which the process of steps S806 to S810 has not been completed for all k. Note that the value at k=0 is determined, for example, by using the results of the heat transfer solidification calculation (temperature T(x, y, t-Δt) and solid fraction f _s (x, y, t-Δt)) at the calculation points 511a, 521a, 531a to 534a on the tracking surface 500a at the heat transfer solidification calculation position n-1 on the upstream side.

If the result of the judgment in step S809 is that the value of k is N (YES in step S809), the process of step S811 is executed. In step S811, the average heat transfer coefficient calculation unit 450 calculates the average heat transfer coefficient _{ht_ave} by performing the calculations of formulas (36a) and (36b) using the intermediate temperature change rate parameter βN _{- 1 calculated in step S808 when k=N. The intermediate temperature change rate parameter βN-1} is calculated at each of two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction among the heat transfer solidification calculation positions n=0 to _nmax . Therefore, the average heat transfer coefficient _{ht_ave} is also calculated at each of two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction among the heat transfer solidification calculation positions n=0 to _nmax .

Next, in step S812, the average external ambient temperature calculation unit 460 calculates the average external ambient temperature T a_ave by executing the calculations of formula (41a) and formula (41b) using the intermediate average external temperature T _b ^* _{_N-1} calculated in step S808 when k = N and the average heat transfer coefficient h _{t_ave} calculated in step S811. The intermediate temperature change rate parameter β _N-1 and the average heat transfer coefficient h _{t_ave} are calculated at each of two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction among _the heat transfer solidification calculation positions n = 0 to n _max . Therefore, the average external ambient temperature T _{a_ave} is also calculated at each of two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction among the heat transfer solidification calculation positions n = 0 to n _max .

Next, in step S813, the heat transfer solidification calculation unit 470 calculates the heat fluxes _qx ( _xB ,y;t), _qy (x, _yB ;t) on the surface of the slab 5 to be used as boundary conditions for the heat transfer solidification calculation for the tracking surface 500a located at the downstream heat transfer solidification calculation position n at time t based on the average heat transfer coefficients _{ht_ave} ( _xB ,y;t), _{ht_ave} (x, _yB ;t) calculated in step S811 and the average external ambient temperatures _T a_ave(xB, _y ;t), T _{a_ave} (x, _yB ;t) calculated in step S812.

The heat transfer solidification calculation unit 470 uses the temperature T(x,y,t-Δt) and solid fraction _fs (x,y,t-Δt) at each calculation point 511a, 521a, 531a to 534a of the tracking surface 500a at the upstream heat transfer solidification calculation position n-1 as initial conditions, and uses the heat fluxes qx( _xB ,y;t), _qy (x,yB;t) on the surface of the slab 5 as boundary conditions in the heat transfer solidification calculation for the tracking surface 500a located at the downstream heat transfer solidification calculation position n to calculate the _temperature T(x,y,t) and solid fraction _fs (x, _y ,t) at each calculation point 511a, 521a, 531a to 534a of the tracking surface 500a at the downstream heat transfer solidification calculation position n. Then, the heat transfer solidification calculation unit 470 uses the result of the heat transfer solidification calculation for the tracking surface 500a at the upstream heat transfer solidification calculation position n-1 used as the initial condition (the result of the heat transfer solidification calculation at time t-Δt) as the initial value for the heat transfer solidification calculation for the tracking surface 500a starting from the downstream heat transfer solidification calculation position n at time t+Δt.

The processing of step S813 is immediately executed for each of the heat transfer solidification calculation positions n=1 to _nmax , and the temperatures T(x, y, t) and solid fractions _fs (x, y, t) at each of the calculation points 511a, 511b, 521a, 521b, 531a to 534a, and 531a to 531b are calculated.

Next, in step S814, the output unit 480 transmits to the control device 100b information including the temperatures T(x,y,t) at the calculation points 511a, 521a, 531a to 534a of the tracking surface 500a after movement to the downstream heat transfer solidification calculation position n, calculated in step S813, including the temperatures T(x,y,t) at the calculation point 511a at the specified position where the target temperature value is set or the calculation points 531a to 534a closest to the specified position.

Next, in step S815, the control unit 490 of the control device 100b determines the amount of cooling water for each cooling zone as an operation amount so that the deviation between the target value and the calculated value of the temperature at a predetermined position of the slab 5 at each heat transfer solidification calculation position n approaches zero. If the predetermined position of the slab 5 coincides with the position of the calculation point 511a indicating the temperature T(x, y, t) transmitted in step S814, the calculated value becomes the temperature T(x, y, t). If the predetermined position of the slab 5 does not coincide with the position of the calculation point 511a indicating the temperature T(x, y, t) transmitted in step S814, the calculated value becomes the temperature at the predetermined position calculated using the temperature T(x, y, t). Then, the control unit 490 sets and outputs the water amount instruction value for each of the cooling sprays 2a to 2t based on the determined amount of cooling water, and adjusts the opening of the flow control valves 3a to 3e.

Next, in step S816, the estimation device 100a determines whether or not to end the processing in the estimation device 100a. For example, when the operator performs an operation on the estimation device 100a to instruct the end of the processing, the estimation device 100a determines that the processing in the estimation device 100a is to be ended. If the result of the determination in step S816 is that the processing in the estimation device 100a is not to be ended (NO in step S816), the processing in step S817 is executed. In step S817, the estimation device 100a updates the time t to time t+Δt. Then, the processing in steps S802 to S816 is executed for the updated time t.
As a result of the determination in step S816, if the processing in the estimation device 100a is to be ended (YES in step S816), the processing in the flowchart of FIG. 8 ends.

(Calculation example)
Next, a calculation example will be described.
In this calculation example, the surface temperature of the slab 5 in the secondary cooling zone of the continuous casting machine was calculated for each of the reference example, the invention example, and the comparative example.
In the reference example, the distance between two adjacent heat transfer and solidification calculation positions in the casting direction (the above-mentioned Δz) is shortened to 5 mm, and the surface temperature of the slab 5 is calculated by heat transfer and solidification calculation, so that a heat transfer and solidification calculation that reflects changes in each cooling method in the secondary cooling zone can be performed. In the reference example, no intermediate calculation positions are set, and the heat transfer and solidification calculation at each heat transfer and solidification calculation position is performed using the result of the heat transfer and solidification calculation at a heat transfer and solidification calculation position adjacent to the heat transfer and solidification calculation position on the upstream side of the heat transfer and solidification calculation position. In the reference example, the calculation accuracy is higher than when Δz is set to 50 mm, for example, but on the other hand, the calculation time is longer.

On the other hand, in the example of the invention, the distance between two adjacent heat transfer solidification calculation positions in the casting direction (the above-mentioned Δz) is set to 50 mm, and nine intermediate calculation positions are set at 5 mm intervals within this distance, and the surface temperature of the slab 5 is calculated using the method of the above-mentioned embodiment. In other words, since the Δz in the example of the invention is 1/10 of the Δz in the reference example, the number of heat transfer solidification calculations, which have a high calculation load, can be reduced to 1/10 in the example of the invention compared to the reference example.

In addition, in order to confirm the accuracy of the temperature calculation according to the invention, in the comparative example, the surface temperature of the slab 5 is calculated as follows. In the comparative example, as in the invention, the interval between two adjacent heat transfer solidification calculation positions in the casting direction is set to 50 mm, and nine intermediate calculation positions are set at 5 mm intervals within this interval. In the comparative example, the heat transfer coefficient at the intermediate calculation position set between two adjacent heat transfer solidification calculation positions in the casting direction is calculated using the result of the heat transfer solidification calculation at the upstream heat transfer solidification calculation position (surface temperature of the slab 5), as in the technology described in Patent Document 2. In the comparative example, the arithmetic mean value of the heat transfer coefficients at each intermediate calculation position calculated in this way is used as the average heat transfer coefficient, and the heat transfer solidification calculation is performed at the downstream heat transfer solidification calculation position to calculate the surface temperature of the slab 5. In the comparative example, the heat transfer coefficient is calculated using the calculated value of the surface temperature of the slab 5 at the upstream heat transfer solidification calculation position at each intermediate calculation position, so that the change in the surface temperature of the slab 5 between the heat transfer solidification calculation positions is not taken into account, as in the invention.

Figure 9 shows the relationship between the width center temperature and the casting length in the reference example and the comparative example. Figure 10 shows the relationship between the width center temperature and the casting length in the reference example and the inventive example. The width center temperature refers to the surface temperature of the slab 5 at the center in the width direction (x-axis direction). The casting length refers to the length in the casting direction with the position of the molten steel meniscus 4 as 0 (zero). Figures 9 and 10 show the width center temperature at each position from the position of the molten steel meniscus 4 to the position of the machine end outlet 9.

As shown in FIG. 9, there is a large difference of about 80°C to 100°C between the width center temperature calculated by the comparative example method and the width center temperature calculated by the reference example method. On the other hand, as shown in FIG. 10, the difference between the width center temperature calculated by the invention example method and the width center temperature calculated by the reference example method cannot be expressed on the same temperature scale as FIG. 9, and the width center temperature calculated by the invention example method cannot be distinguished from the width center temperature calculated by the reference example method, even though Δz is 10 times longer than that of the reference example method. FIG. 11 is an enlarged view of a portion of FIG. 10. In the invention example method, although the interval between two adjacent heat transfer solidification calculation positions in the casting direction is larger than that of the reference example method, as shown in FIG. 11, the invention example and the reference example are generally consistent overall. In this way, the method of the present invention achieves accuracy close to that of the reference example method, while reducing the calculation time to nearly 1/10 of that of the reference example method. Therefore, in the example of the invention, it is possible to complete the heat transfer solidification calculation at the downstream heat transfer solidification calculation position while casting progresses for the distance between two adjacent heat transfer solidification calculation positions in the casting direction, with the accuracy required for practical use.

(summary)
As described above, in this embodiment, the estimation device 100a executes a heat transfer calculation for the slab surface at the intermediate calculation position k set between the heat transfer and solidification calculation positions n-1 and n using the result of the heat transfer and solidification calculation for the tracking surface 500a at the upstream heat transfer and solidification calculation position n-1 of the two heat transfer and solidification calculation positions n-1 and n adjacent to each other in the casting direction, and executes a heat transfer and solidification calculation for the slab surface at the intermediate calculation position k set between the heat transfer and solidification calculation positions n-1 and n using the result of the heat transfer calculation. Therefore, even if the interval between the tracking faces for executing the heat transfer and solidification calculation is not shortened, the heat transfer and solidification calculation for the tracking surface 500a at the downstream heat transfer and solidification calculation position n can be executed in consideration of the temperature change of the surface of the slab 5 and its neighboring area between the two heat transfer and solidification calculation positions n-1 and n adjacent to each other in the casting direction. Therefore, it is possible to suppress both the decrease in the estimation accuracy of the temperature and solid fraction of the slab 5 cast by the continuous casting machine and the increase in the calculation load.

In the present embodiment, the estimation device 100a calculates an average heat transfer coefficient h t_ave, which is an average value of the heat transfer coefficient between the slab 5 and the outside between the upstream heat transfer solidification calculation position n-1 and the downstream heat transfer solidification calculation position n, using the result of the heat transfer calculation at time t for an intermediate calculation position k set between two heat transfer solidification calculation positions n and n-1 adjacent to each other in the casting direction. The estimation device 100a also calculates an average external atmosphere temperature _{T a_ave} , which is an average value of the external atmosphere temperature T a of the slab 5 between the upstream heat transfer solidification calculation position n-1 and the downstream heat transfer solidification calculation position n, using the result of the heat transfer calculation at time t for an intermediate calculation position _k set between two heat transfer solidification calculation positions adjacent to each other in the casting direction and the average heat transfer coefficient h _{t_ave} . The estimation device 100a then calculates the heat fluxes q _{x (x B , y; t), q y (x, y B ; t) (i.e., the heat fluxes q x (x B , y} ; t), q _y (x, y _B ; t) at each time t on the surface of the slab 5 until the tracking surface 500a moves from the upstream heat transfer solidification calculation position n-1 to the downstream heat transfer solidification calculation position n) based on the average heat transfer _coefficient h _{t_ave} and the average external atmosphere temperature T _{a_ave} , and performs the heat transfer solidification calculation at the downstream heat transfer solidification calculation position n at time _t using the heat fluxes q _x , q _y at time t on the surface of the slab 5. Therefore, it is possible to perform the heat transfer solidification calculation when the tracking surface 500a is located at the downstream heat transfer solidification calculation position n, taking into account the change in the heat transfer coefficient h _t between two heat transfer solidification calculation positions adjacent in the casting direction. Therefore, even if a region such as region d, where the heat transfer coefficient _ht is large and the heat fluxes _qx , _qy change nonlinearly with temperature, is included between two heat transfer solidification calculation positions adjacent in the casting direction, it is possible to suppress a decrease in the estimation accuracy of the temperature and solid fraction of the slab 5 cast by the continuous casting machine.

In addition, in this embodiment, the estimation device 100a performs heat transfer calculations for the surface of the slab 5 at time t for an intermediate calculation position k set between two adjacent heat transfer and solidification calculation positions n-1 and n in the casting direction as one-dimensional heat transfer calculations in a direction perpendicular to the surface of the slab 5, and performs heat transfer and solidification calculations for the downstream heat transfer and solidification calculation position n at time t as two-dimensional heat transfer and solidification calculations in directions parallel to and perpendicular to the surface of the slab 5. Therefore, the calculation load of the heat transfer calculation can be reduced more than the calculation load of the heat transfer and solidification calculation.

In addition, in this embodiment, the estimation device 100a sets multiple intermediate calculation positions k between two heat transfer solidification calculation positions n-1 and n adjacent to each other in the casting direction, and performs heat transfer calculation for the surface of the slab 5 at time t for each intermediate calculation position k using the result of the heat transfer solidification calculation at time t-Δt at the upstream heat transfer solidification calculation position n-1 and the result of the heat transfer calculation for the surface of the slab 5 at time t for an intermediate calculation position upstream of the intermediate calculation position k. Therefore, many intermediate calculation positions k can be set between two heat transfer solidification calculation positions n-1 and n adjacent to each other in the casting direction. Therefore, it is possible to perform heat transfer solidification calculation for the tracking surface 500a at the downstream heat transfer solidification calculation position n, more accurately reflecting the temperature change on the surface of the slab 5 and its neighboring area between the two heat transfer solidification calculation positions n-1 and n adjacent to each other in the casting direction.

In the present embodiment, the estimation device 100a calculates the heat transfer coefficient h _{t_k-1} and the external ambient temperature T _{a_k-1} at the intermediate calculation position k-1, calculates the intermediate external temperature T _b ^ _{_k-1} at the intermediate calculation position k-1 using the calculated results, and calculates the intermediate average external temperature T _b ^* _{_k-1} and the intermediate temperature change rate parameter β _k-1 at the intermediate calculation position k-1 using the calculated results, and repeats this process by updating k from 1 to N. Therefore, the temperature change of the surface of the slab 5 and its surrounding area between two adjacent heat transfer solidification calculation positions n-1 and n in the casting direction can be quantitatively expressed using the intermediate average external temperature T _b ^* _{_k-1} , and the change in the ease of heat and temperature transfer between the slab 5 and the outside between two adjacent heat transfer solidification calculation positions n-1 and n in the casting direction can be quantitatively expressed using the intermediate temperature change rate parameter β _k-1 . Therefore, the deterioration of the estimation accuracy of the temperature and solid fraction of the slab 5 can be further suppressed.

Furthermore, in this embodiment, the estimation device 100a calculates the intermediate surface temperature T^k at the intermediate calculation position k using the calculation results of the heat transfer coefficient h _{t_k-1} and the external ambient temperature T _{a_k-1} at the intermediate calculation position k-1, and calculates the heat transfer coefficient h _{t_k-1} at the intermediate calculation position k-1 using the intermediate surface temperature T^ _k-1 at the intermediate calculation position k. Therefore, the heat transfer coefficient h _{t ,} which changes depending on the temperature, can be quantitatively obtained.

In addition, in this embodiment, the intermediate calculation position k is set so that the cooling method for the surface of the slab 5 at at least one of the two heat transfer solidification calculation positions n-1 and n adjacent in the casting direction in the secondary cooling zone is different from the cooling method for the surface of the slab 5 at at least one intermediate calculation position k set between the two heat transfer solidification calculation positions n-1 and n. Therefore, even if there are multiple cooling method areas between the heat transfer solidification calculation position n-1 and the heat transfer solidification calculation position n, the temperature change of the surface of the slab 5 and the area nearby in the area between the two heat transfer solidification calculation positions n-1 and n can be estimated based on the result of the heat transfer calculation at the intermediate calculation position k. Therefore, the deterioration of the estimation accuracy of the temperature and solid fraction of the slab 5 can be further suppressed.

In this embodiment, the interval in the casting direction between two intermediate calculation positions k adjacent in the casting direction is set to 1/2 or less of the assumed minimum value of the length in the casting direction of the region d where the mist-like cooling water sprayed from the cooling sprays 2a to 2t collides with the slab 5. Therefore, a plurality of intermediate calculation positions k are located in the region d. Therefore, the temperature change of the surface of the slab 5 and its neighboring region in the region where the value of the heat transfer coefficient _ht is large and the heat fluxes _qx , _qy change nonlinearly with respect to temperature, such as the region d, between two heat transfer and solidification calculation positions n-1 and n adjacent in the casting direction, can be estimated based on the results of the heat transfer calculation at the plurality of intermediate calculation positions k. Therefore, it is possible to further suppress the decrease in the estimation accuracy of the temperature and solid fraction of the slab 5 when the region d is included between two heat transfer and solidification calculation positions n-1 and n adjacent in the casting direction.

In the present embodiment, the estimation device 100a executes a heat transfer solidification calculation for the entire surface of the tracking surface 500a, and therefore can calculate the distribution of the temperature T(x, y, t) and the solid fraction _fs (x, y, t) of the tracking surface 500a with higher accuracy.
Furthermore, in this embodiment, the estimation device 100a performs heat transfer calculations with the rectangular region 710 of the surface layer, the rectangular region 720 of the second layer adjacent to the rectangular region 710 of the surface layer on the inside of the slab 5, and the outer rectangular region 730 adjacent to the rectangular region 710 of the surface layer on the outside of the slab 5 as regions to be transferred. This makes it possible to limit the range affected by heat transfer on the surface of the slab 5. This makes it possible to further reduce the calculation load of the heat transfer calculation on the surface of the slab 5.

In addition, in this embodiment, the control device 100b controls the operation of the flow control valves 3a to 3e (the amount of cooling water in the secondary cooling zone of the continuous casting machine) so that the deviation between the target value and the calculated value of the temperature at a predetermined position (e.g., the surface or center) of the slab 5 being cast by the continuous casting machine approaches zero based on the results of the heat transfer solidification calculation between the heat transfer solidification calculation positions n-1 and n executed by the estimation device 100a. Therefore, it is possible to prevent cracks from occurring on the surface of the slab 5, central segregation from occurring in the slab 5, and solidification from not being completed to the center of the slab 5 at the machine end outlet 9.

(Modification)
In the present embodiment, the continuous casting control device 10 (system) including the estimation device 100a and the control device 100b as independent devices has been described as an example. However, this is not necessarily required. The continuous casting control device may be configured as a single device including the functions of the estimation device 100a and the functions of the control device 100b.

In the present embodiment, the heat fluxes q _x and q _y at the surface of the slab ₅ at time t are calculated as boundary conditions in the heat transfer and solidification calculation based on the average heat transfer coefficient h _{t_ave} and the average external atmosphere temperature T a_ave. However, this is not necessarily required as long as the boundary conditions in the heat transfer and solidification calculation are set using the result of the heat transfer calculation for the intermediate calculation position k. For example, the result of the heat transfer and solidification calculation at the upstream heat transfer and solidification calculation position n-1 is used as a starting point, and the heat transfer coefficient and the external atmosphere temperature at each intermediate calculation position k are calculated using the result of the heat transfer calculation for the intermediate calculation position k while shifting the intermediate calculation position k to the downstream side, thereby calculating the boundary conditions in the heat transfer and solidification calculation.

The above-described embodiments of the present invention can be realized by a computer executing a program. A computer-readable recording medium on which the program is recorded and a computer program product such as the program can also be applied as embodiments of the present invention. Examples of recording media that can be used include flexible disks, hard disks, optical disks, magneto-optical disks, CD-ROMs, magnetic tapes, non-volatile memory cards, and ROMs. The embodiments of the present invention can also be realized by a PLC (Programmable Logic Controller) or dedicated hardware such as an ASIC (Application Specific Integrated Circuit).

Furthermore, the above-described embodiments of the present invention are merely examples of the implementation of the present invention, and the technical scope of the present invention should not be interpreted in a limiting manner based on these. In other words, the present invention can be implemented in various forms without departing from its technical concept or main features.

1 Mold 2a to 2t Cooling spray 3a to 3e Flow rate control valve 4 Molten steel meniscus 5 Cast piece 5a Narrow side surface 5b to 5c Long side surface 6a to 6f Cooling zone boundary line 7a to 7v Support roll 8a to 8p Support roll 9 End outlet 100 Continuous casting control device 100a Estimation device 100b Control device 410 Operation data acquisition unit 420 Heat transfer and solidification calculation position setting unit 430 Intermediate calculation position setting unit 440 Heat transfer calculation unit 441 First calculation unit 442 Second calculation unit 443 Third calculation unit 450 Average heat transfer coefficient calculation unit 460 Average external ambient temperature calculation unit 470 Heat transfer and solidification calculation unit 480 Output unit 490 Control unit 500a to 500e Tracking surface 510a to 510b Rectangular area of surface layer 511a, 512a Calculation points 520a to 520b Rectangular area of second layer 520a, 521b Calculation points 531a to 534a Calculation points 531b to 534b Calculation points 600 Cross section for heat transfer calculation 710 Rectangular area of surface layer 711 Calculation point 720 Rectangular area of second layer 721 Calculation point 730 Outer rectangular area 731 Calculation point a to e Areas with different cooling methods n-1 to n Heat transfer solidification calculation position k Intermediate calculation position Δx _c Length in the x-axis direction from the boundary line of the cross section for heat transfer calculation to the calculation point of the second layer Δy _c Length in the y-axis direction from the boundary line of the cross section for heat transfer calculation to the calculation point of the second layer Δx _s Length in the x-axis direction of the rectangular areas of the surface layer aligned in the y-axis direction Δy _s Length in the y-axis direction of the rectangular areas of the surface layer aligned in the x-axis direction

Claims (14)

An estimation device for estimating a state of a slab cast by a continuous casting machine,
a heat transfer and solidification calculation unit that performs heat transfer and solidification calculation for a tracking surface, which is a cross section of the slab perpendicular to the casting direction, at a plurality of heat transfer and solidification calculation positions that are set at intervals in the casting direction;
a heat transfer calculation unit that performs a heat transfer calculation for a surface of the slab at at least one intermediate calculation position that is set between two of the heat transfer and solidification calculation positions that are adjacent to each other in the casting direction;
Equipped with
the heat transfer calculation unit executes a heat transfer calculation for the surface of the slab at the intermediate calculation position by using a result of a heat transfer and solidification calculation for the tracking surface at an upstream heat transfer and solidification calculation position of the two heat transfer and solidification calculation positions adjacent to each other in the casting direction;
The heat transfer solidification calculation unit uses the result of the heat transfer calculation for the surface of the slab at the intermediate calculation position to perform a heat transfer solidification calculation for the tracking surface when the tracking surface at the upstream heat transfer solidification calculation position is located at the downstream heat transfer solidification calculation position of two heat transfer solidification calculation positions adjacent to each other in the casting direction. an average heat transfer coefficient calculation unit that calculates an average heat transfer coefficient, which is an average value of the heat transfer coefficient between the slab and the outside during the period from the upstream heat transfer solidification calculation position to the downstream heat transfer solidification calculation position, using a result of heat transfer calculation for the surface of the slab at the intermediate calculation position;
an average external atmosphere temperature calculation unit that calculates an average external atmosphere temperature, which is an average value of the external atmosphere temperature of the slab during the period when the tracking surface moves from the upstream heat transfer solidification calculation position to the downstream heat transfer solidification calculation position, using a result of heat transfer calculation for the surface of the slab at the intermediate calculation position and the average heat transfer coefficient;
Furthermore,
2. The estimation device according to claim 1, wherein the heat transfer solidification calculation unit calculates a heat flux on a surface of the slab until the tracking surface at the upstream heat transfer solidification calculation position moves to the downstream heat transfer solidification calculation position based on the average heat transfer coefficient and the average external atmosphere temperature, and performs a heat transfer solidification calculation for the tracking surface at the upstream heat transfer solidification calculation position when the tracking surface is positioned at the downstream heat transfer solidification calculation position, using the heat flux on the surface of the slab at the downstream heat transfer solidification calculation position. the heat transfer calculation unit executes, as a heat transfer calculation for the surface of the slab, a one-dimensional heat transfer calculation for a direction along a short side surface when the surface is a long side surface, and a one-dimensional heat transfer calculation for a direction along a long side surface when the surface is a short side surface ;
The estimation device according to claim 1 , wherein the heat transfer solidification calculation unit executes a two-dimensional heat transfer solidification calculation for the tracking surface as the heat transfer solidification calculation. A plurality of intermediate calculation positions are set between two adjacent heat transfer solidification calculation positions in the casting direction;
The estimation device according to any one of claims 1 to 3, wherein the heat transfer calculation unit executes a heat transfer calculation for the surface of the slab at the intermediate calculation position by using a result of a heat transfer and solidification calculation for the tracking surface at the upstream heat transfer and solidification calculation position and a result of a heat transfer calculation for the intermediate calculation position upstream of the intermediate calculation position. The heat transfer calculation unit is
A first calculation unit that calculates a heat transfer coefficient between the slab and an outside and an outside atmosphere temperature of the slab;
A second calculation unit that calculates an intermediate external temperature using a result of the calculation in the first calculation unit;
a third calculation unit that calculates an intermediate average external temperature and an intermediate temperature change rate parameter using the results of the calculation in the second calculation unit;
having
the calculation in the first calculation unit, the calculation in the second calculation unit, and the calculation in the third calculation unit are repeatedly executed by updating the intermediate calculation position to be calculated in order from the intermediate calculation position on the upstream side;
The intermediate external temperature is a temperature determined based on temperatures inside and outside the slab at an intermediate calculation position that is one position upstream of the intermediate calculation position to be calculated,
The intermediate average external temperature represents an average value of the intermediate external temperature between the upstream heat transfer solidification calculation position and the intermediate calculation position to be calculated,
The estimation device according to any one of claims 1 to 4, wherein the intermediate temperature change rate parameter is a parameter representing an average value of ease of heat and temperature transfer between the slab and the outside from the upstream heat transfer solidification calculation position to the intermediate calculation position to be calculated. The second calculation unit further calculates an intermediate surface temperature, which is a surface temperature of the slab, using a result of the calculation in the first calculation unit;
The estimation device according to claim 5 , wherein the first calculation unit calculates the heat transfer coefficient between the slab and an outside by using the intermediate surface temperature at the intermediate calculation position that is one position upstream of the intermediate calculation position to be calculated. The estimation device estimates a state of a slab cooled in a secondary cooling zone of the continuous casting machine,
The estimation device according to any one of claims 1 to 6, wherein the intermediate calculation positions are set such that a cooling method for the surface of the slab at at least one of the two heat transfer solidification calculation positions adjacent to each other in the casting direction in the secondary cooling zone is different from a cooling method for the surface of the slab at at least one of the intermediate calculation positions set between the two heat transfer solidification calculation positions. The estimation device estimates a state of a slab cooled in a secondary cooling zone of the continuous casting machine,
A plurality of intermediate calculation positions are set between the two heat transfer solidification calculation positions adjacent to each other in the casting direction,
The estimation device according to any one of claims 1 to 7, wherein a distance in the casting direction between two of the intermediate calculation positions adjacent to each other in the casting direction is equal to or less than half of a value assumed as a minimum value of a length in the casting direction of a region where cooling water collides with the slab in the secondary cooling zone. The estimation device according to any one of claims 1 to 8, wherein the heat transfer solidification calculation unit performs heat transfer solidification calculation for the entire surface of the tracking surface. The estimation device according to any one of claims 1 to 9, wherein the heat transfer calculation unit performs heat transfer calculations at the intermediate calculation position, with a discretized region including the surface of the slab and discretized regions adjacent to the discretized region on the inside and outside of the slab as target regions for heat transfer. An estimation device according to any one of claims 1 to 10;
a control device that controls the amount of cooling water supplied to the slab so that a deviation between a target value and a calculated value of a temperature at a predetermined position of the slab being cast by the continuous casting machine approaches zero based on a result of the heat transfer and solidification calculation for the tracking surface executed by the estimation device;
A continuous casting control device comprising: A method for estimating a state of a slab cast by a continuous casting machine, comprising:
a heat transfer and solidification calculation step of performing a heat transfer and solidification calculation for a tracking surface, which is a cross section of the slab perpendicular to the casting direction, at a plurality of heat transfer and solidification calculation positions set at intervals in the casting direction;
a heat transfer calculation step of performing a heat transfer calculation for a surface of the slab at at least one intermediate calculation position set between two of the heat transfer and solidification calculation positions adjacent to each other in a casting direction;
having
The heat transfer calculation step performs a heat transfer calculation for a surface of the slab at the intermediate calculation position using a result of a heat transfer and solidification calculation for a tracking surface at an upstream heat transfer and solidification calculation position of the two heat transfer and solidification calculation positions adjacent to each other in the casting direction,
The heat transfer and solidification calculation step is an estimation method in which, using a result of a heat transfer calculation for the surface of the slab at the intermediate calculation position, a heat transfer and solidification calculation is performed for the tracking surface when the tracking surface at the upstream heat transfer and solidification calculation position is located at the downstream heat transfer and solidification calculation position of two heat transfer and solidification calculation positions adjacent to each other in the casting direction. A continuous casting control method comprising a control step of controlling the amount of cooling water supplied to the slab so that the deviation between the target value and the calculated value of the temperature at a predetermined position of the slab being cast by the continuous casting machine approaches zero based on the results of the heat transfer and solidification calculation for the tracking surface performed by the estimation method described in claim 12. A program for causing a computer to function as each part of the estimation device according to any one of claims 1 to 10.

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