Method for on-line heat status tracking of a cast strand

Abstract

The application provides a kind of casting billet online thermal state tracking method, comprising the following steps: according to the rule of block division, the asymmetric profiled blank is divided into area block, and the casting billet center area block and the casting billet surface area block are obtained respectively;The structured coarse grid division is carried out to the casting billet center area block, and the structured fine grid division is carried out to the casting billet surface area block, to obtain a plurality of structured grids, and the grid nodes of the structured grid are indexed and described, and the grid data of each structured block and the index are stored correspondingly;The heat transfer equation coefficient matrix and the solidification latent heat release source term of the grid unit of each structured block are calculated;The temperature field of the grid unit in the structured block is calculated by coupling the incomplete matrix LU decomposition and SIP algorithm in turn. By using the application, the problems of the prior art, such as unsuitable for asymmetric profiled blank and slab dual-purpose casting machine, reduced system reliability and time step limitation, can be solved.

Description

Technical Field

[0001] This invention relates to the field of continuous casting technology, and more specifically, to a method for tracking the online thermal state of a cast billet. Background Technology

[0002] For irregularly shaped continuous casting billets with complex cross-sectional shapes, structured orthogonal meshes cannot be used for temperature field calculations. Previously, the inventors of this technology proposed a method for billet thermal tracking using an explicit discontinuous Galerkin finite element method, obtaining an invention patent (patent number, ZL201811231595.6). In a steel plant's practical application, a quarter-shaped continuous casting billet was used, and a triangular finite element mesh was employed for billet temperature field calculations. Online thermal tracking of irregularly shaped continuous casting billets was first implemented in China. At that time, production control could be performed on two-strand irregularly shaped billets. Later, the steel plant expanded its capacity, increasing from two casting strands to three; however, the secondary process control computer was not upgraded, still using the original secondary control software. Only by modifying the configuration file could it adapt to the tasks of online thermal tracking and dynamic water distribution for three-strand irregularly shaped billets. However, the CPU resources consumed by numerical calculations increased, and the system redundancy did not exceed 60%. Although no failures have occurred so far, the system's reliability has been reduced.

[0003] Another feature of the continuous casting machine in this steel plant is that it is a dual-purpose casting machine for both slabs and irregularly shaped billets. For the secondary process control software, it needs to be able to calculate the temperature field of both slabs and irregularly shaped billets simultaneously. At the time, two different methods were used: for slabs, a structured orthogonal mesh was used with the finite volume method for numerical calculation; for irregularly shaped billets, an unstructured mesh was used with the finite element method. The finite element method calculation program itself is relatively complex, and the addition of the finite volume method further complicates the secondary process control software, making it difficult to maintain and upgrade later. With increasingly fierce competition in the steel industry, multi-purpose machines have become a feasible way to reduce costs. From this perspective, it is necessary to develop a new, unified method for billet thermal tracking.

[0004] Furthermore, the plant proposed using asymmetric shaped billets to roll urban rail (Skyrail). To this end, our company designed a continuous casting machine and developed a production process for the plant. Considering the geometry and cooling state of the asymmetric shaped billets, the original online thermal tracking system for shaped continuous casting billets is no longer usable. The main reasons are twofold: First, the computational domain is expanded, requiring at least half of the cross-section instead of 1 / 4. Second, the cooling of asymmetric shaped billets is more uneven, making them prone to cracking, necessitating better control of secondary cooling. This requires finer meshing of the billet. For explicit algorithms, the computational time step is limited by stability conditions; finer meshing significantly reduces computational efficiency because the time step is inversely proportional to the square of the mesh size. Therefore, careful analysis is needed regarding the appropriate computational mesh and temperature field calculation method for asymmetric shaped billets.

[0005] Considering the slabs and billets commonly used in steel production, whose geometry is symmetrical, and assuming the cooling conditions are also symmetrical, a quarter section can be used for analysis. Currently, for these continuously cast billets, most online temperature field calculation models generally employ the explicit finite volume method, using a quarter section and a structured orthogonal mesh for calculation. This calculation method is relatively simple. For large-section slab billets, especially some special steel grades, surface and corner cracks are prone to occur during production. Considering the non-uniformity of spray cooling on the billet surface, a finer mesh is needed to carefully characterize the billet cooling boundary conditions. This requires temperature field calculations for the entire billet to better control its cooling. Compared to earlier models, the computational domain is four times larger. Furthermore, if a finer mesh is used near the billet surface and corners, the calculation time step to satisfy the stability conditions will be smaller, significantly impacting the computational efficiency of earlier models.

[0006] Therefore, in summary, the existing online thermal state tracking methods for billets are not suitable for use in casting machines that can handle both asymmetrical billets and slabs. Furthermore, the mesh generation methods used in the existing technology cannot specifically consider the internal heat transfer distribution of asymmetrical billets, resulting in reduced system reliability. In addition, the explicit finite volume method used in the existing technology to calculate the temperature field of continuously cast billets is limited by the time step. Summary of the Invention

[0007] In view of the above problems, the purpose of this invention is to provide an online thermal state tracking method for cast billets, so as to solve the problems that the existing online thermal state tracking methods for cast billets are not suitable for use in casting machines that can handle both asymmetrical billets and slabs. Furthermore, the mesh generation method used in the prior art cannot specifically consider the internal heat transfer distribution of asymmetrical billets, resulting in reduced system reliability. In addition, the explicit finite volume method used in the prior art is subject to time step limitations when calculating the temperature field of continuously cast billets.

[0008] This invention provides a method for online thermal state tracking of cast billets, comprising the following steps:

[0009] S1. According to the preset block division rules, the asymmetrical irregular billet is divided into blocks to obtain the central block and the surface block of the billet.

[0010] S2. Perform structured coarse mesh division on the central region block of the billet, and perform structured fine mesh division on the surface region block of the billet to obtain multiple structured meshes. Then, index and describe the nodes of the structured meshes, and store the mesh data and index of each structured mesh accordingly.

[0011] S3. Based on the grid data and index, calculate the heat transfer equation coefficient matrix and solidification latent heat release source term for each structured grid in the central region block of the billet and each structured grid in the surface region block of the billet, respectively.

[0012] S4. Perform incomplete matrix LU decomposition on the coefficient matrix of the heat transfer equation for the central region block and the surface region block of the billet, respectively, and calculate the residuals in combination with the latent heat release source term to obtain the residual vector of the structured mesh in the central region block and the residual vector of the structured mesh in the surface region block of the billet, respectively.

[0013] S5. Based on the residual vector of the structured mesh in the central region block of the billet and the residual vector of the structured mesh in the surface region block of the billet, the temperature field of the central region block and the surface region block of the billet is coupled and calculated using the SIP algorithm to obtain the temperature field of the structured mesh of multiple blocks, and to ensure thermal balance at the interface of the region blocks. The temperature field that meets the preset convergence condition is obtained through iterative algorithm.

[0014] S6. Add the temperature value of each structured grid obtained according to the temperature field that meets the preset convergence condition to the grid data, and return to step S3 according to the preset time step, advance the time, and store the temperature value of each structured grid at different times and its corresponding structured grid index.

[0015] Furthermore, a preferred embodiment is that the block division rules include: ensuring that the asymmetrical irregular billet maintains a uniform thickness in the direction from the surface of the billet to the interior of the billet, and that the boundary of the central area block of the billet is enclosed by a broken line or a circular arc curve segment.

[0016] Furthermore, a preferred embodiment involves performing structured coarse mesh division on the central region block of the cast billet and structured fine mesh division on the surface region block of the cast billet to obtain multiple structured meshes. The nodes of these multiple structured meshes are then indexed and described. The mesh data and index of each structured mesh are stored accordingly, including:

[0017] The central region of the billet is divided into a coarse grid according to a first preset grid size, and the surface region of the billet is divided into a fine grid according to a second preset grid size, resulting in multiple structured grids.

[0018] The position coordinates of the nodes of the structured grid are marked to determine the position of the structured grid by means of index description, and the grid data and index of each structured grid are stored accordingly.

[0019] In addition, a preferred approach is to store the structured mesh data obtained after fine mesh division of the surface region block of the billet according to an O-type or C-type data structure, and to establish a correspondence between the structured meshes on both sides of the boundary between the central region block of the billet and the surface region block of the billet, and to store the correspondence.

[0020] Furthermore, a preferred embodiment is that the calculation formula for the latent heat release source term is:

[0021] in,

[0022] ρ is the density of the cast billet, g l S is the liquidus fraction, L is the latent heat of solidification, t is time, T is temperature; P S is the proportional term after linearization of the latent heat release source term in heat transfer calculations. C S is a constant term. P and S C These are all solidification characteristic parameters of steel grades.

[0023] Furthermore, a preferred embodiment further includes, before calculating the latent heat release source terms for each structured grid within the central region block of the billet and each structured grid within the surface region block of the billet, the following:

[0024] Establish a data table of solidification characteristic parameters of steel grades corresponding to each billet temperature within a preset temperature range; when calculating the latent heat release source item, obtain the solidification characteristic parameters corresponding to the current billet temperature by looking up the data table.

[0025] Furthermore, a preferred approach is to calculate the heat transfer equation coefficient matrix for each structured grid within the central region block of the cast billet and for each structured grid within the surface region block of the cast billet, including:

[0026] A set of heat transfer discrete equations is determined based on the heat transfer discrete equations of the internal nodes of the structured mesh that does not contain the boundary between the central region block and the surface region block of the slab, and the heat transfer discrete equations of the internal boundary that contain the boundary between the central region block and the surface region block of the slab.

[0027] Based on the aforementioned discrete heat transfer equations, determine the coefficient matrix of the heat transfer equations.

[0028] Furthermore, in a preferred embodiment, the heat transfer discretization equation for the internal nodes of the structured mesh that does not include the boundary between the central region block and the surface region block of the billet is:

[0029] in,

[0030] The subscript l represents the index of an adjacent structured grid, and the value of l ranges from the four adjacent structured grids: East (E), West (W), South (S), and North (N). The subscript P represents the current structured grid. P A is a coefficient. l The matrix coefficients of the structured grids adjacent to the current structured grid P are φ, where φ is the solution variable; Q p For load vectors;

[0031] The internal boundary heat transfer discretization equation, which includes the boundary between the central region block and the surface region block of the billet, is as follows:

[0032] For the central region of the cast billet, the internal boundary heat transfer discretization equation is:

[0033]

[0034] For the surface region of the cast billet, the internal boundary heat transfer discretization equation is:

[0035] in,

[0036] P is a structured mesh containing the internal boundaries of the central region block and the surface region block of the billet. The subscript l takes values ​​from adjacent cells that do not involve the internal boundaries, with values ​​of West (W), South (S), and North (N). Subscripts L and R represent the left and right adjacent cells on both sides of the internal boundaries, respectively. Array A R and A L These are the coefficients of the equation.

[0037] Furthermore, a preferred embodiment is that the formula for the incomplete matrix LU decomposition of the coefficient matrix of the heat transfer equation of the structured grid is:

[0038] Aφ=b;

[0039] LU=A+N=M; where A is the coefficient matrix of the heat transfer equation, M is an approximate matrix of A, N is the difference matrix between M and A, and N satisfies the condition Nφ≈0.

[0040] Furthermore, a preferred approach is to use the SIP algorithm to coupledly calculate the temperature fields of the central region block and the surface region block of the slab based on the residual vectors of the structured mesh within the central region block and the residual vectors of the structured mesh within the surface region block of the slab, thereby obtaining the temperature fields of multiple structured meshes and ensuring thermal equilibrium at the interface of the region blocks. In the process of obtaining the temperature field that satisfies the preset convergence condition through an iterative algorithm,

[0041] The formula for calculating the temperature field based on the residual vector iteration is: (A+N)δ n+1 =r n ;in,

[0042] δ n+1 r is the correction value for the temperature field. n Let A be the residual vector of the nth calculation, A be the coefficient matrix of the heat transfer equation, and N be the difference matrix.

[0043] The formula for the SIP algorithm is: (LU)δ n+1 =r n .

[0044] As can be seen from the above technical solution, the online thermal state tracking method for billets provided by this invention solves the problem of meshing complex-shaped billets by dividing the asymmetrical irregular billet into a central region block and a surface region block, and then performing structured coarse meshing on the central region block and structured fine meshing on the surface region block. This results in a high-quality structured mesh. The use of a fine mesh in the area near the billet surface where heat transfer is intense, and a coarser mesh in the interior, improves computational efficiency. This mesh is applicable to all continuous casting billet shapes. The developed billet thermal tracking technology meets the requirements of multi-functional continuous casting machines and reduces the maintenance and upgrade costs of secondary control software. The implicit finite volume method is used to calculate the temperature field of the continuous casting billet, solving the problem of time step limitations in explicit methods. For special steel slabs and asymmetrical irregular billets, where finer meshing, more detailed description of heat transfer and solidification, and stricter control of uniform cooling are required, the method provided by this invention is more advantageous.

[0045] To achieve the foregoing and related objectives, one or more aspects of the invention include the features that will be described in detail below. The following description and accompanying drawings illustrate certain exemplary aspects of the invention. However, these aspects indicate only a few of the various ways in which the principles of the invention can be used. Furthermore, the invention is intended to encompass all such aspects and their equivalents. Attached Figure Description

[0046] Other objects and results of the invention will become more apparent and readily understood with reference to the following description taken in conjunction with the accompanying drawings. In the drawings:

[0047] Figure 1 This is a flowchart illustrating the online thermal state tracking method for cast billets according to an embodiment of the present invention.

[0048] Figure 2 This is a schematic diagram of the cross-sectional mesh division of an asymmetric irregular-shaped billet according to an embodiment of the present invention;

[0049] Figure 3 This is a schematic diagram of slab mesh division according to an embodiment of the present invention;

[0050] Figure 4 This is a schematic diagram of a finite volume computational grid for a computational unit in a region block according to an embodiment of the present invention;

[0051] Figure 5 This is a schematic diagram of the finite volume computation mesh of the internal boundary cells on the left and right sides of the boundary between two region blocks according to an embodiment of the present invention.

[0052] Figure 6 This is a schematic diagram of an unstructured FEM (finite element method) computational mesh according to Embodiment 1 of the present invention;

[0053] Figure 7 This is a schematic diagram of a structured Multi-blocks FVM (Finite Volume Machine) mesh according to Embodiment 1 of the present invention;

[0054] Figure 8 This is a comparison diagram of the crystallizer outlet temperature cloud map according to Embodiment 1 of the present invention. The left side is FEM and the right side is FVM.

[0055] Figure 9 This is a comparison diagram of the end temperatures of the two cooling zones according to Embodiment 1 of the present invention. The left side is FEM and the right side is FVM.

[0056] Figure 10 This is a comparison diagram of the cooling curves of the billet according to Embodiment 1 of the present invention.

[0057] In the accompanying drawings, the same reference numerals indicate similar or corresponding features or functions. Detailed Implementation

[0058] In the following description, numerous specific details are set forth for illustrative purposes and to provide a thorough understanding of one or more embodiments. However, it will be apparent that these embodiments may also be implemented without these specific details.

[0059] To address the issues that the aforementioned online thermal state tracking method for billets is unsuitable for casting machines that can handle both asymmetrical billets and slabs, and that the mesh generation method used in the prior art cannot specifically consider the internal heat transfer distribution of asymmetrical billets, leading to reduced system reliability; and that the explicit finite volume method used in the prior art is subject to time step limitations in calculating the temperature field of continuously cast billets, a new online thermal state tracking method for billets is proposed.

[0060] The specific embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0061] To illustrate the online thermal state tracking method for cast billets provided by this invention, Figure 1 The flowchart of the online thermal state tracking method for cast billets according to an embodiment of the present invention is shown; Figure 2 The mesh division of the cross-section of the asymmetric irregular billet according to an embodiment of the present invention is shown; Figure 3 The slab mesh division according to an embodiment of the present invention is shown; Figure 4 A finite-volume computational grid for computational cells in a region block according to an embodiment of the present invention is shown. Figure 5 A finite volume computational mesh for an internal boundary cell in the boundary between two region blocks according to an embodiment of the present invention is shown. Figure 6 An unstructured FEM (finite element method) computational mesh according to Embodiment 1 of the present invention is shown; Figure 7 A structured Multi-blocks FVM (Multi-block Finite Volume) mesh according to Embodiment 1 of the present invention is shown; Figure 8 A comparison of crystallizer outlet temperature cloud maps according to Embodiment 1 of the present invention is shown, with FEM on the left and FVM on the right; Figure 9 The diagram shows a comparison of the end temperatures of the two cooling zones according to Embodiment 1 of the present invention, with FEM on the left and FVM on the right. Figure 10 The comparison of the billet cooling curves according to Embodiment 1 of the present invention is shown, with the FEM result plotted using a thick solid line and the FVM result plotted using a thin solid line.

[0062] like Figures 1 to 10 As shown in the figure, the online thermal state tracking method for cast billets provided by the present invention includes the following steps:

[0063] S1. According to the preset block division rules, the asymmetrical irregular billet is divided into blocks to obtain the central block and the surface block of the billet.

[0064] As a preferred embodiment of the present invention, the block division rules include: ensuring that the thickness of the asymmetrical irregular billet remains uniform from the surface of the billet to the interior of the billet, and that the boundary of the central area block of the billet is enclosed by curve segments such as broken lines or circular arcs.

[0065] Taking the most complex asymmetrical irregular blank as an example, we will explain how to generate a mesh. Figure 2 As shown, the cross-section of this irregularly shaped billet is symmetrical from left to right but asymmetrical from top to bottom; half of it can be taken as the object of study. The cross-sectional dimensions of the billet are hidden, and only the cross-section is drawn to scale. For clarity, a coarse grid is used to divide the billet cross-section. Figure 2 This is a multi-block mesh. If the graphic is cut along the centerline AB of the web portion and the BCDEFB boundary line of the flange center portion, removing the area enclosed by BCDEFB in the flange center portion, and then straightening the remaining part, a logically structured mesh (Block A) can be obtained. This type of mesh is called a C-type mesh. Another region is a different structured mesh (Block B). This meshing method yields a structured mesh.

[0066] To illustrate the versatility of the technology used, Figure 3 A complete example of slab cross-section mesh generation is provided. The region near the slab surface constitutes Block A, which logically corresponds to the toroidal mesh generation and is referred to as O-type; the central part of the slab is Block B. It should be noted that the developed finite volume temperature field calculation program is universal, capable of handling both non-orthogonal and orthogonal meshes; it can handle single blocks as well as multiple blocks. For slabs and billets, a single orthogonal structured mesh can be used, similar to earlier programs.

[0067] S2. Perform structured coarse mesh division on the central region block of the billet and structured fine mesh division on the surface region block of the billet to obtain multiple structured meshes. Index and describe the nodes of the structured meshes and store the mesh data and index of each structured mesh accordingly.

[0068] As a preferred embodiment of the present invention, the central region of the cast billet is divided into a structured coarse mesh, and the surface region of the cast billet is divided into a structured fine mesh, resulting in multiple structured meshes. The nodes of these multiple structured meshes are indexed and described, and the mesh data and index of each structured mesh are stored accordingly, including:

[0069] The central region of the billet is divided into a coarse grid according to the first preset grid size, and the surface region of the billet is divided into a fine grid according to the second preset grid size, resulting in multiple structured grids.

[0070] The position coordinates of the grid nodes of the structured grid are labeled to determine the position of the structured grid by means of index description, and the grid data and index of each structured grid are stored accordingly.

[0071] As a preferred embodiment of the present invention, the grid data of the structured grid obtained after fine grid division of the surface region block of the billet is stored according to the O-type or C-type data structure, and the correspondence between the structured grids on both sides of the boundary between the central region block of the billet and the surface region block of the billet is established and stored.

[0072] For each region, the nodes of Block A and Block B are described using indices (i,j), making grid data storage very convenient. In addition, there are two obvious advantages. First, it is easy to control the mesh density and rationally divide the mesh. For example, near the surface of the billet, where temperature changes are greater, a finer mesh is used; while inside the billet, where the temperature gradient is smaller, a coarser mesh can be used. The selection principle (or the selection principle of the BCDEFB closed curve) for Block A is to maintain a uniform thickness as much as possible in the direction from the billet surface inwards, consistent with the solidification law of the billet. The division principle for Block B is not limited to selecting a closed curve formed by broken lines; arcs and other curve segments can also be selected. Especially for example... Figure 2 As shown, in segments CD, FE, and CBF, appropriate circular arc curves are selected to make the resulting mesh closer to an orthogonal mesh, or in technical terms, with a smaller skewness. In short, this meshing method yields a higher-quality computational mesh, controls the scale of the entire heat transfer discrete control equation, and optimizes and improves computational speed. Secondly, it facilitates the display of calculation results and data post-processing. For example, to display the surface temperature distribution of an irregularly shaped billet and analyze the uniformity of billet cooling, using a C-type mesh on the billet surface is convenient; simply extract the temperature values ​​from the first layer of the mesh and plot the graphs and curves according to the mesh coordinates. To display the temperature distribution a few millimeters below the billet surface, simply extract the temperature data from the corresponding layer according to the pre-divided mesh. However, using an unstructured finite element mesh is less convenient, requiring interpolation calculations.

[0073] S3. Based on the grid data and index, calculate the heat transfer equation coefficient matrix and solidification latent heat release source term for each structured grid in the central region block of the billet and each structured grid in the surface region block of the billet.

[0074] As a preferred embodiment of the present invention, the calculation formula for the latent heat release source term is as follows:

[0075] in,

[0076] ρ is the density of the cast billet, g l S is the liquidus fraction, L is the latent heat of solidification, t is time, T is temperature; P S is the proportional term after linearization of the latent heat release source term in heat transfer calculations. C S is a constant term. P and S C These are all solidification characteristic parameters of steel grades.

[0077] Since the temperature gradient in the billet pulling direction (Z-axis direction) is very small, its thermal conductivity effect can be ignored relative to the heat transfer caused by the billet movement. Only the thermal conductivity in the other two directions (X and Y) of the billet cross-section needs to be considered. If the heat transfer effect caused by the molten steel flow is handled by amplifying the thermal conductivity of the liquid metal, the commonly used heat transfer control equation for the thin-plate moving method is:

[0078]

[0079] In the above formula, κ eff It is the equivalent thermal conductivity, which is taken as mκ for liquids in the turbulent region. l , κ l It is the thermal conductivity of molten steel, κ l Take a constant, where m is the magnification factor, determined empirically; for the solidified shell, the thermal conductivity is the solid-phase thermal conductivity κ. s The thermal conductivity of the two-phase region varies with temperature. eff =(1-g l )κ s +g l κ l Calculate g l C represents the liquid phase fraction. p Specific heat. Similar to the treatment of thermal conductivity, the specific heat of a two-phase mixture is determined by C. p =(1-g l C ps +g l C pl Calculate, where C ps and C pl These are the specific heats of the solid and liquid phases, respectively. The second term on the right-hand side of equation (1) represents the latent heat release caused by the change in the liquid phase fraction over time.

[0080] A common method for dealing with latent heat of solidification is to linearize the source term. As a source term in equation (1), the liquid fraction g must be known in order to perform the calculation. lThe variation of g with temperature T, assuming g l It has a linear relationship with T:

[0081]

[0082] In the formula T s and T l These are the solidus and liquidus temperatures, respectively. Substituting equation (2) into equation (1), the source term can be written as:

[0083]

[0084]

[0085]

[0086] In the above formula, T 0 Δt and Δt represent the temperature and calculation time step of the previous moment, respectively. Note that a linear processing method was used in the discretization of the latent heat source term, and the proportionality coefficient in equation (3) is negative. This ensures that the diagonal elements dominate in the coefficient matrix of the discrete equation system, satisfying the positive definite condition of the equation.

[0087] As a preferred embodiment of the present invention, before calculating the latent heat release source terms of each structured grid in the central region block of the billet and each structured grid in the surface region block of the billet, the method further includes:

[0088] Establish a data table of solidification characteristic parameters of steel grades corresponding to the temperature of each billet within a preset temperature range;

[0089] When calculating the latent heat release source term, the solidification characteristic parameters corresponding to the current billet temperature are obtained by looking up data tables. During the calculation process, the solidification parameters Sp and Sc are quickly looked up using the table number index.

[0090] Assuming a linear relationship between liquidus fraction and temperature, a method for handling latent heat of solidification is presented. However, in reality, the relationship between them is not linear during the solidification process of the cast billet. Based on the solidification characteristics of steel grades produced by continuous casting, we can assume a piecewise linear relationship between liquidus fraction and temperature, and the above method can be used for each segment. The release of latent heat of solidification introduces strong nonlinear characteristics into the heat transfer equation. If latent heat of solidification is not handled properly, it will cause difficulties in solving the implicit equation system, or even failure. Piecewise linearization of the liquidus fraction versus temperature curve can effectively handle the release of latent heat of solidification. However, piecewise linearization requires finding the temperature interval and knowing which solidification interval the current temperature belongs to. If the segment interval needs to be determined in each iteration, and then the slope and other parameters of the line segment need to be calculated, it will be very wasteful of CPU time. To speed up the calculation, an integer is allocated to each calculation node to record the position index i of the solidification interval segment. Referring to formula (3), the constants such as Sp and Sc on each segment are calculated in advance according to the solidification law of steel grade, and a data table is built. It can be used directly in subsequent calculations to avoid repeated calculations.

[0091] To reduce the CPU time spent searching for the solidification range, the characteristics of continuous casting billet solidification can be utilized. Generally, the billet temperature gradually decreases over time; this characteristic can be used to optimize the search process. Assume temperature T... * (T i <T * ≤T i+1 The corresponding liquid phase ratio is gl * (gl i <gl * ≤gl i+1 This belongs to the i-th segment of the solidification curve. To find T... * The corresponding gl * If the data table is segmented, and T is searched in a loop starting from the beginning... * If the segment belongs to a certain range, then i comparison operations are required, which is very wasteful. We can use the position index variable i of the solidification interval segment, if the temperature T' of the previous moment (or previous iteration) of the computation node has already been recorded. * At the corresponding position i' on the relationship curve, first determine the current temperature T. * Is it still in segment i' (T)? i' <T * ≤T i '+1 If the conditions are met, the data table can be directly referenced, which is a common situation in calculations. If the conditions are not met, comparisons are made preferentially in the direction of decreasing temperature (T). i'-1 <T * ≤T i'If the condition of temperature decrease is not met, the search proceeds in the opposite direction. This allows for table lookup with only a few judgments. Pre-calculating the solidification parameter data table (Sp, Sc, and T relationship table) and using the position index i of the solidification interval segments, along with an optimized route for parameter lookup, can significantly improve computational efficiency.

[0092] As a preferred embodiment of the present invention, the heat transfer equation coefficient matrix is ​​calculated for each structured grid in the central region block of the cast billet and for each structured grid in the surface region block of the cast billet, including:

[0093] The heat transfer discrete equation set is determined based on the heat transfer discrete equations of the internal nodes of the structured mesh that does not contain the boundary between the central region block and the surface region block of the slab, and the heat transfer discrete equations of the internal boundary that does contain the boundary between the central region block and the surface region block of the slab.

[0094] Determine the coefficient matrix of the heat transfer equations based on the discrete heat transfer equation set.

[0095] As a preferred embodiment of the present invention, the heat transfer discretization equation for the internal nodes of the structured mesh that does not include the boundary between the central region block and the surface region block of the billet is:

[0096] in,

[0097] The subscript l represents the index of an adjacent structured grid, and the value of l ranges from the four adjacent structured grids: East (E), West (W), South (S), and North (N). The subscript P represents the current structured grid. P A is a coefficient. l The matrix coefficients of the structured grids adjacent to the current structured grid P are φ, where φ is the solution variable; Q p For load vectors;

[0098] The internal boundary heat transfer discretization equation, including the boundary between the central region block and the surface region block of the billet, is as follows:

[0099] For the central region of the billet, the internal boundary heat transfer discretization equation is:

[0100]

[0101] For the surface region of the cast billet, the internal boundary heat transfer discretization equation is:

[0102] in,

[0103] P is a structured mesh containing the internal boundaries of the central region block and the surface region block of the billet. The subscript l takes values ​​from adjacent cells that do not involve the internal boundaries, with values ​​of West (W), South (S), and North (N). Subscripts L and R represent the left and right adjacent cells on both sides of the internal boundaries, respectively. Array A R and A L These are the coefficients of the equation.

[0104] like Figure 4 Combination Figure 5 As shown, in the finite volume method, the parameter φ at the element interface must be calculated. e Its gradient value can be approximated using linear interpolation. For example, the parameter φ at point e on the interface. e Calculate according to the following formula:

[0105] φ e ≈φ e' =φ E λ e +φ P (1-λ e )

[0106] Wherein, the interpolation coefficient λ e The calculation is based on the geometric parameters according to the following formula, where r e r E and r P It is a geometric point vector:

[0107]

[0108] When the computational mesh is severely skewed, there is a significant difference between points e and e', and the parameter φ at point e on the interface... e The following formula needs to be used for correction calculation:

[0109] φ e ≈φ e' +(gradφ) e' ·(r e -r e' )

[0110] To avoid numerical oscillations, the gradient at point e on the interface is calculated using the following formula:

[0111]

[0112] in Taking the calculation result from the previous time step, this term appears as a source term in the discrete equations. Based on the finite volume method, the discrete equations can be derived using the variable φ on the element interface and its gradient parameters:

[0113]

[0114] For convenience, instead of using E, W, S, and N for mesh numbering on block interfaces, use L and R for numbering, record adjacency relationships, and use A. R and A L Replace the original A E The coefficients of the discrete equations are represented by numbers and stored in a special array. A R and A L The calculation method and A E The calculation methods for the coefficients of discrete equations are the same.

[0115] S4. Perform incomplete matrix LU decomposition on the coefficient matrices of the heat transfer equations for the central region block and the surface region block of the billet, respectively, and calculate the residuals by combining the latent heat release source terms of solidification, to obtain the residual vectors of the structured mesh in the central region block and the surface region block of the billet, respectively.

[0116] As a preferred embodiment of the present invention, the formula for the incomplete matrix LU decomposition of the coefficient matrix of the heat transfer equation of the structured grid is as follows:

[0117] Aφ=b;

[0118] LU=A+N=M; where A is the coefficient matrix of the heat transfer equation, M is an approximate matrix of A, N is the difference matrix between M and A, and N satisfies the condition Nφ≈0.

[0119] S5. Based on the residual vectors of the structured mesh in the central region block of the billet and the residual vectors of the structured mesh in the surface region block of the billet, the SIP algorithm is used to couple and calculate the temperature fields of the central region block and the surface region block of the billet, so as to obtain the temperature fields of the structured mesh of multiple blocks and ensure the thermal balance at the interface of the region blocks. The temperature field that meets the preset convergence condition is obtained through iterative algorithm.

[0120] As a preferred embodiment of the present invention, the temperature fields of the central region block and the surface region block of the billet are coupled and calculated using the SIP algorithm based on the residual vectors of the structured mesh in the central region block and the residual vectors of the structured mesh in the surface region block of the billet. This yields the temperature fields of multiple structured meshes while ensuring thermal equilibrium at the interface of the region blocks. The process involves obtaining a temperature field that satisfies a preset convergence condition through an iterative algorithm.

[0121] The formula for calculating the temperature field based on the residual vector iteration is: (A+N)δ n+1 =r n ;in,

[0122] δ n+1 r is the correction value for the temperature field. nLet A be the residual vector of the nth calculation, A be the coefficient matrix of the heat transfer equation, and N be the difference matrix.

[0123] The formula for the SIP algorithm is: (LU)δ n+1 =r n .

[0124] Specifically, the iterative equations for solving the temperature field on a structured grid using the SIP algorithm include:

[0125] (A+N)φ=(A+N)φ+(b-Aφ)

[0126] When the equation yields a convergent solution, (b-Aφ)=0, therefore, the following iterative method can be used:

[0127] (A+N)φ n+1 =(A+N)φ n +(b-Aφ n );Right now:

[0128]

[0129] (A+N)δ n+1 =r n

[0130] φ n+1 and φ n It is the result of the (n+1)th and nth iterations, and the difference δ n+1 It is an intermediate variable in the calculation process, and a correction factor in the iterative calculation. n It is the residual vector calculated in the nth time.

[0131] According to the SIP algorithm:

[0132] (LU)δ n+1 =r n

[0133] Uδ n+1 =L -1 r n =R n

[0134] Correction amount δ n+1 Calculate according to the following formula:

[0135] δ n+1 =U -1 R n

[0136] The SIP algorithm makes full use of the characteristics of the upper diagonal strip matrix, resulting in high computational efficiency.

[0137] Besides simplifying data storage, using structured grids also simplifies computation. For example, regarding the interface between two adjacent computational cells i and i+1 along the i-direction, which is the western interface for cell i and the eastern interface for cell i+1, the heat flux across this interface is equal for both cells, differing only by a negative sign. Therefore, regarding the coefficients of the discrete equations, the coefficient A of cell i... W The coefficient A of unit i+1 E The difference is only a negative sign, requiring only one calculation. Furthermore, to reduce CPU computation, the interface interpolation calculation employs a loop similar to the SIMPLE method in computational fluid dynamics. Here's a brief explanation: within one time step, there are two loops: an outer loop and an inner loop. In the outer loop, the coefficient matrix of the discrete equations is generated, and an incomplete LU decomposition is performed on the coefficient matrix. In the inner loop, the temperature field is solved through SIP iteration. Because the LU matrix decomposition performed in the outer loop remains unchanged in the inner loop, requiring only one LU decomposition, the computational efficiency is very high.

[0138] S6. Add the temperature value of each structured grid obtained from the temperature field that meets the preset convergence condition to the grid data, and return to step S3 according to the preset time step to advance the time and store the temperature value of each structured grid at different times and its corresponding structured grid index.

[0139] The following examples will further illustrate the present invention so that those skilled in the art can better understand its advantages and features.

[0140] Example 1:

[0141] To demonstrate the reliability of the method provided by this invention, program verification was first performed using a simple example with analytical solutions. An analytical solution to an unsteady semi-infinite one-dimensional heat conduction problem is given: with a uniform initial temperature field, starting from time 0, boundary conditions of types 1, 2, and 3 are applied respectively, and the change of internal temperature over time is calculated. The analytical solution is a transcendental function composed of error functions. The numerical calculation results are consistent with the theoretical solution, and the plotted curves coincide. Calculations show that a mesh size of 2 to 3 mm is sufficient to meet the accuracy requirements of continuous casting engineering.

[0142] To verify the calculation using a multi-block non-orthogonal mesh, a heat conduction example of a thick-walled circular tube in steady-state one-dimensional cylindrical coordinates was used. If the inner and outer surface temperatures of the tube remain constant, the temperature distribution inside the tube follows the natural logarithm. The computational domain was artificially divided into two annexes, and each region was partitioned using an O-type mesh to form a multi-block structured mesh. The numerical calculations were consistent with the theoretical solutions. Due to space limitations, the results of these simple examples are not presented.

[0143] The following is based on Figure 6 The following example of a continuously cast asymmetric irregular-shaped billet is used for application illustration. The parameters used in the calculation are very close to the actual working conditions. The billet's physical properties are as follows: thermal conductivity 30 (W / m²). -1 .℃ -1 ), specific heat 650.0 (J·kg) -1 .℃ -1 Density 7200.0 (Kg·m³) -3 Latent heat of solidification 309000 (J·kg) -1 The liquidus temperature is 1520°C, and the solidus temperature is 1480.0°C. The cooling zone division and overall heat exchange coefficient are shown in Table 1. The continuous casting production speed is 0.6 m / min, and the casting temperature is 1550°C.

[0144]

[0145] Table 1

[0146] Based on a comparison of simple examples with analytical solutions and numerical methods, and considering the fact that a 2-3 mm mesh is sufficient for engineering accuracy requirements, a finer (slightly less than 3 mm) uniform triangular mesh was used as the benchmark for comparison in finite element calculations. Figure 6 As shown, there are a total of 15,242 nodes and 29,823 elements. Due to the large number of computational meshes, the computational workload is very large when using the method of directly solving the system of equations; therefore, it takes approximately 7 hours to complete a 30-minute continuous casting billet simulation calculation on a desktop computer with an i9 CPU.

[0147] If the casting billet is divided into the following portions according to the method provided by this invention: Figure 7 The mesh shown is as follows: Block A is the surface layer of the slab, close to the surface, with a minimum mesh size of 2mm in the thickness direction. The mesh gradually thickens towards the interior of the slab in a proportional manner, with the thickest mesh approximately 6mm, totaling 130×15 meshes. The scaling factor for thickening the mesh ranges from 1.03 to 1.08. Block B is the center of the slab flange, divided into 6×30 meshes, with a maximum mesh size of approximately 12mm. The total number of mesh elements is 2130 (130×15 + 6×30 = 2130). Compared to FEM calculations, the number of meshes is significantly reduced, by two orders of magnitude. The same calculation takes only 4 minutes of CPU time.

[0148] To verify the calculation results, the temperature fields at the crystallizer outlet (0.7m from the meniscus) and the secondary cooling zone outlet (10.06m from the meniscus) were plotted and compared. Figure 8 and Figure 9 As shown, the temperature distribution is consistent.

[0149] like Figure 10As shown, four characteristic points are selected: the center of the web surface (point A), the large radius (point B), the center of the side surface (point C), and the final solidification hot spot (point D). The cooling curve of the cast billet is plotted for further comparison. (See attached image.) Figure 10 Because the two grid points are not completely identical, there are some differences in the cooling curves of the solidification hot node (point D), due to the coarser FVM grid; the data for the other points overlap. Using the FVM grid as described above, ideal calculation results can be obtained, the computation CPU time is significantly reduced, and the requirements for online temperature field calculations can be met.

[0150] As can be seen from the above specific embodiments, the online thermal state tracking method for billets provided by the present invention solves the problem of meshing complex-shaped billets by dividing asymmetrical irregular billets into central region blocks and surface region blocks, and then performing structured coarse meshing on the central region blocks and structured fine meshing on the surface region blocks. This results in a high-quality structured mesh. The use of a fine mesh in the area near the billet surface where heat transfer is intense, and a coarser mesh in the interior, improves computational efficiency. This mesh is applicable to all continuous casting billet shapes. The developed billet thermal tracking technology meets the requirements of multi-functional continuous casting machines and reduces the maintenance and upgrade costs of secondary control software. The implicit finite volume method is used to calculate the temperature field of the continuous casting billet, solving the problem of time step limitations in explicit methods. For special steel slabs and asymmetrical irregular billets, where finer meshing, more detailed description of heat transfer and solidification, and stricter control of uniform cooling are required, the method provided by the present invention is more advantageous.

[0151] The online thermal condition tracking method for cast billets according to the present invention has been described above by way of example with reference to the accompanying drawings. However, those skilled in the art should understand that various modifications can be made to the online thermal condition tracking method for cast billets proposed above without departing from the scope of the present invention. Therefore, the scope of protection of the present invention should be determined by the contents of the appended claims.

Claims

1. A method for online thermal state tracking of cast billets, characterized in that, Includes the following steps: S1. According to the preset block division rules, the asymmetrical irregular billet is divided into blocks to obtain the central block and the surface block of the billet. S2. Perform structured coarse mesh division on the central region block of the billet, and perform structured fine mesh division on the surface region block of the billet to obtain multiple structured meshes. Then, index and describe the nodes of the structured meshes, and store the mesh data and index of each structured mesh accordingly. S3. Based on the grid data and index, calculate the heat transfer equation coefficient matrix and solidification latent heat release source term for each structured grid in the central region block of the billet and each structured grid in the surface region block of the billet, respectively. S4. Perform incomplete matrix LU decomposition on the coefficient matrix of the heat transfer equation for the central region block and the surface region block of the billet, respectively, and calculate the residuals in combination with the latent heat release source term to obtain the residual vector of the structured mesh in the central region block and the residual vector of the structured mesh in the surface region block of the billet, respectively. S5. Based on the residual vector of the structured mesh in the central region block of the billet and the residual vector of the structured mesh in the surface region block of the billet, the temperature field of the central region block and the surface region block of the billet is coupled and calculated using the SIP algorithm to obtain the temperature field of the structured mesh of multiple blocks, and to ensure thermal balance at the interface of the region blocks. The temperature field that meets the preset convergence condition is obtained through iterative algorithm. S6. Add the temperature value of each structured grid obtained according to the temperature field that meets the preset convergence condition to the grid data, and return to step S3 according to the preset time step, advance the time, and store the temperature value of each structured grid at different times and its corresponding structured grid index.

2. The online thermal state tracking method for cast billets according to claim 1, characterized in that, The block division rules include: The asymmetrical shaped billet maintains a uniform thickness from the surface of the billet towards the interior of the billet, and the boundary of the central region of the billet is enclosed by a broken line or a circular arc curve segment.

3. The online thermal state tracking method for cast billets according to claim 1, characterized in that, The process involves dividing the central region of the cast billet into a coarse, structured mesh, and dividing the surface region of the cast billet into a fine, structured mesh, resulting in multiple structured meshes. The nodes of these structured meshes are then indexed and described. The mesh data and index of each structured mesh are stored accordingly. The central region of the billet is divided into a coarse grid according to a first preset grid size, and the surface region of the billet is divided into a fine grid according to a second preset grid size, resulting in multiple structured grids. The position coordinates of the nodes of the structured grid are marked to determine the position of the structured grid by means of index description, and the grid data and index of each structured grid are stored accordingly.

4. The online thermal state tracking method for cast billets according to claim 3, characterized in that, The grid data of the structured grid obtained after fine grid division of the surface region block of the billet is stored according to the O-type or C-type data structure, and the correspondence between the structured grids on both sides of the boundary between the central region block of the billet and the surface region block of the billet is established and stored.

5. The online thermal state tracking method for cast billets according to claim 1, characterized in that, The formula for calculating the latent heat release source term is as follows: in, ρ is the density of the cast billet, g l S is the liquidus fraction, L is the latent heat of solidification, t is time, T is temperature; P S is the proportional term after linearization of the latent heat release source term in heat transfer calculations. c S is a constant term. P and S C These are all solidification characteristic parameters of steel grades.

6. The online thermal state tracking method for cast billets according to claim 5, characterized in that, Before calculating the latent heat release source terms for each structured grid within the central region block of the cast billet and each structured grid within the surface region block of the cast billet, the method further includes: Establish a data table of solidification characteristic parameters of steel grades corresponding to each billet temperature within a preset temperature range; when calculating the latent heat release source item, obtain the solidification characteristic parameters corresponding to the current billet temperature by looking up the data table.

7. The online thermal state tracking method for cast billets according to claim 3, characterized in that, The heat transfer equation coefficient matrix is ​​calculated for each structured grid within the central region block of the cast billet and for each structured grid within the surface region block of the cast billet, including: A set of heat transfer discrete equations is determined based on the heat transfer discrete equations of the internal nodes of the structured mesh that does not contain the boundary between the central region block and the surface region block of the slab, and the heat transfer discrete equations of the internal boundary that does contain the boundary between the central region block and the surface region block of the slab. Based on the aforementioned discrete heat transfer equations, determine the coefficient matrix of the heat transfer equations.

8. The online thermal state tracking method for cast billets according to claim 7, characterized in that, The heat transfer discretization equation for the internal nodes of the structured mesh that does not include the boundary between the central region block and the surface region block of the billet is: in, The subscript l represents the index of an adjacent structured grid, and the value of l ranges from the four adjacent structured grids: East (E), West (W), South (S), and North (N). The subscript P represents the current structured grid. P A is a coefficient. l The matrix coefficients of the structured grids adjacent to the current structured grid P are φ, where φ is the solution variable; Q p For load vectors; The internal boundary heat transfer discretization equation, which includes the boundary between the central region block and the surface region block of the billet, is as follows: For the central region of the cast billet, the internal boundary heat transfer discretization equation is: For the surface region of the cast billet, the internal boundary heat transfer discretization equation is: in, P is a structured mesh containing the internal boundaries of the central region block and the surface region block of the billet. The subscript l takes values ​​from adjacent cells that do not involve the internal boundaries, with values ​​of West (W), South (S), and North (N). Subscripts L and R represent the left and right adjacent cells on both sides of the internal boundaries, respectively. Array A R and A L These are the coefficients of the equation.

9. The online thermal state tracking method for cast billets according to claim 8, characterized in that, The formula for the incomplete matrix LU decomposition of the coefficient matrix of the heat transfer equation of the structured grid is: Aφ=b; LU=A+N=M; where A is the coefficient matrix of the heat transfer equation, M is an approximate matrix of A, N is the difference matrix between M and A, and N satisfies the condition Nφ≈0.

10. The online thermal state tracking method for cast billets according to claim 9, characterized in that, In the process of calculating the temperature fields of the central region block and the surface region block of the slab using the SIP algorithm based on the residual vectors of the structured mesh within the central region block and the residual vectors of the structured mesh within the surface region block, the temperature fields of multiple structured meshes are obtained, ensuring thermal equilibrium at the interface of the region blocks. An iterative algorithm is then used to obtain the temperature field that satisfies the preset convergence condition. The formula for calculating the temperature field based on the residual vector iteration is: (A+N)δ n+1 =r n ;in, δ n+1 r is the correction value for the temperature field. n Let A be the residual vector of the nth calculation, A be the coefficient matrix of the heat transfer equation, and N be the difference matrix. The formula for the SIP algorithm is: (LU)δ n+1 =r n .

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