A finite volume calculation method for solidification heat transfer of irregular cross-section continuous casting billet

Abstract

This invention designs a finite volume method for calculating the solidification heat transfer of irregular cross-section continuously cast billets. First, a 1 / 2 continuous casting billet simulation domain is constructed based on the cross-sectional dimensions of the irregular cross-section billet. An unstructured mesh is then created within this domain, and the mesh information is saved. Next, an unstructured finite volume heat transfer model is established to address the solidification heat transfer problem during continuous casting. Finally, this model is used to simulate the solidification heat transfer phenomenon of irregular continuously cast billets during the continuous casting process. The volume method is used for calculation, which improves the calculation speed compared to the finite element method. Appropriate boundary conditions are selected based on different cold zones to more accurately simulate the solidification heat transfer throughout the entire continuous casting process of irregular continuously cast billets. The method fully considers the actual situation of continuous casting of irregular billets, describes the temperature field changes during the continuous casting process, and thus provides support for the uniform cooling of irregular continuously cast billets and the improvement of their quality.

Description

Technical Field

[0001] This invention relates to the field of steel continuous casting technology, and in particular to a finite volume calculation method for solidification heat transfer of irregular cross-section continuous casting billets. Background Technology

[0002] Continuous casting is a process in which molten steel is formed into a billet with a specific shape and size after steelmaking. During continuous casting, molten steel flows from the tundish into the crystallizer and first exchanges heat with the copper wall, causing a portion of the steel to solidify and form a primary billet shell. Under the action of a straightening machine, the billet continues to descend into the secondary cooling zone for final cooling.

[0003] A square billet is a steel billet with a square cross-section produced by a continuous casting machine. Large square billet continuous casting machines are mainly used to produce medium and high carbon alloy steels for rolling high-strength profiles, wire rods, channel steels, and other steel grades with high requirements for internal quality and compression ratio. A reasonable secondary cooling system can ensure the uniform and stable cooling process of the billet, preventing the formation of quality defects such as cracks and bulging. At the same time, the shape and depth of the liquid phase cavity in the billet have a very important influence.

[0004] Continuous casting of round billets has the advantages of high billet precision, good quality, low energy consumption, and high metal yield. However, continuously cast round billets still have many quality defects, mainly internal and surface defects. Internal quality defects in continuously cast round billets mainly include interdendritic cracks and central shrinkage cavities. Surface defects are mainly surface cracks, whose morphologies include longitudinal cracks, star-shaped cracks, surface dents, and hairline cracks.

[0005] Shaped billets refer to continuously cast billets with complex cross-sections, other than square, slab, round, and rectangular billets. Shaped billets are widely used in transportation, construction, and heavy equipment manufacturing due to their excellent load-bearing capacity, sectional stability, and weight savings. However, because of their complex cross-sectional shapes, shaped billets are more prone to various quality defects compared to conventionally shaped billets. Furthermore, current research on shaped billet continuous casting technology is insufficient to meet the needs of modern shaped billet continuous casting development, and the level of continuous casting control is still far from meeting expectations.

[0006] In actual production, the corners of square billets are often rounded, the boundaries of round billets are arc-shaped, and the cross-sections of irregular billets are complex and contain multiple arc segments. However, existing technologies usually simulate this by dividing orthogonal mesh units, which results in a sawtooth pattern at the boundaries, greatly reducing the calculation accuracy.

[0007] Currently, there are three main methods for computer numerical simulation: the finite difference method, the finite element method, and the finite volume method. The finite difference method was the earliest method used in computer numerical simulation. This method divides the solution domain into a difference grid, replacing the continuous solution domain with a finite number of grid nodes. Using methods such as Taylor series expansion, the finite difference method discretizes the derivatives in the governing equations by replacing them with the difference quotients of function values ​​at the grid nodes, thus establishing a system of algebraic equations with the values ​​at the grid nodes as unknowns. This method is an approximate numerical solution that directly transforms a differential problem into an algebraic problem. The mathematical concepts are intuitive, the expression is simple, and the mathematical modeling is convenient, making it easy to program and parallelize. However, it is not suitable for handling complex boundaries, and the handling of irregular regions is very cumbersome.

[0008] The basic idea of ​​the finite element method (FEM) is to divide the computational domain into a finite number of non-overlapping elements. Within each element, suitable nodes are selected as interpolation points for the solution function. The variables in the differential equation are rewritten as linear expressions composed of the nodal values ​​of each variable or its derivative and the selected interpolation function. The differential equation is then discretized and solved using variational principles or the weighted residual method. Existing techniques disclose the establishment of a two-dimensional finite element model to geometrically discretize the two-dimensional cross-section of the continuously cast billet. Then, for the heat transfer problem during continuous casting solidification, a two-dimensional finite element solidification heat transfer model of the continuously cast billet is constructed. Finally, information such as the structural parameters of the continuous casting machine is imported, and the model is used to calculate the temperature field changes during the continuous casting process. The use of hybrid triangular and quadrilateral elements can better simulate curved boundaries and adapt to various complex shapes. However, the resulting finite element matrix equation is quite complex. When there are n element nodes, an n×n matrix equation is formed, and the matrix is ​​irregular, making it difficult to solve and resulting in slow convergence speed and low computational efficiency. Secondly, the finite element method has high requirements for mesh quality, and when the mesh quality is poor, there is a high risk of no solution. Summary of the Invention

[0009] To address the shortcomings of existing technologies, this invention provides a finite volume calculation method for the solidification heat transfer of irregular cross-section continuous casting billets. The aim is to calculate the temperature changes at various times during the solidification process of irregular cross-section continuous casting billets, advance the simulation and heat transfer analysis of irregular cross-section continuous casting billets, establish an online solidification heat transfer model, optimize the cooling water distribution in the secondary cooling zone, reduce quality defects in continuous casting billets, and provide a theoretical basis for improving the quality of continuous casting billets.

[0010] A finite volume calculation method for solidification heat transfer of irregular cross-section continuously cast billets, specifically including the following steps:

[0011] Step 1: Construct a 1 / 2 continuous casting billet simulation calculation domain based on the cross-sectional dimensions of the irregular cross-section continuous casting billet; the irregular cross-section continuous casting billet includes shaped billets, rounded square billets, and round billets;

[0012] Based on the cross-sectional dimensions, the coordinates of key points are determined and imported into ANSYS software. In ANSYS, the key points are connected to generate geometric surfaces, constructing a 1 / 2 simulation computational domain. The cross-sectional dimensions for irregularly shaped billets include flange thickness, wide face width, narrow face width, web length, web thickness, inner edge radius, flange radius, and fillet radius; for rounded square billets, they include wide face width, narrow face width, and fillet radius; and for round billets, they include the round billet radius. The key points for irregularly shaped billets include web endpoints, flange endpoints, inner edge radius endpoints, flange endpoints, and fillet endpoints; for rounded square billets, they include fillet endpoints, wide face endpoints, and narrow face endpoints; and for round billets, they include the center point.

[0013] Step 2: Divide the unstructured mesh in the 1 / 2 continuous casting billet simulation calculation domain established in Step 1, and save the mesh information; the mesh information includes element information, node information, and element surface information;

[0014] Step 2.1: Select the element type and element shape in ANSYS, and determine the element size. Then, generate a non-mechanical mesh in the continuous casting billet simulation calculation domain.

[0015] Step 2.2: Output the cell information and node information as a DAT format file and save it; the cell information includes the cell number and which nodes are contained in the cell; the node information includes the node number and node coordinates;

[0016] Step 2.3: Extract and save the element surface information based on the element information and node information;

[0017] Because the cross-sectional shape of the irregular continuous casting billet is relatively complex, ANSYS cannot use quadrilateral elements for meshing when performing region partitioning. There are some triangular elements. Therefore, it is necessary to determine whether all the elements are quadrilateral elements. If not, the mesh needs to be re-partitioned and proceed to step 2.1.

[0018] Extracting element surface information based on the relationship between elements, nodes, and faces, outputting it as a DAT format file and saving it; the relationship between elements, nodes, and faces is that an element consists of four nodes and has four element faces, and each element face consists of two nodes;

[0019] Step 3: To address the heat transfer problem during the solidification of continuously cast billets, a non-structural finite volume heat transfer model is established.

[0020] Step 3.1: Discretize the diffusion equations in the regular Cartesian coordinate system and construct a finite volume calculation model for the steady-state heat transfer problem;

[0021] Step 3.1.1: Discrete steady-state heat transfer and diffusion equations;

[0022] The fundamental variable in the heat transfer process is temperature, which is a function of the geometric position within the object and time. First, consider a regular Cartesian mesh over a simple rectangular domain. Based on this mesh, the steady-state diffusion equation is:

[0023]

[0024] in It represents a scalar quantity, which is temperature in heat transfer problems; It is the diffusion coefficient, which is the thermal conductivity coefficient in heat transfer problems, W / (m·K); Represents the volume per unit volume within this computational domain The generation rate, i.e., the source term, is the internal heat source in heat transfer problems, W / The equation can be written in its general form as follows:

[0025]

[0026] in It is the diffusion flux;

[0027] Taking unit C as an example, equation (1) is applied to unit C. Discretize it and transform it into:

[0028]

[0029] Subscript Representative Unit Each unit surface on, Represents the surface vector. Representative unit C The generation rate, i.e., the source term, Representative Unit Controlling volume, Expanding the above equation, we get:

[0030]

[0031] Subscript , , , The four faces of element C represent the cardinal directions (north, south, east, and west). For a uniform Cartesian mesh, the surface vector of an element face is calculated using the following formula:

[0032]

[0033] in represent Increment in the axial direction, represent Increment in the axial direction, represent Unit vector along the axis, represent Unit vector along the axis;

[0034] Therefore, as an example, the diffusion flux of the eastern unit surface is:

[0035]

[0036] In the discretization of the conservation equation, if a single integration point approach is used, the discrete form of the diffusion flux is:

[0037]

[0038] Assumption If the element C exhibits a linear variation between its centroids, then the element surface... upper edge direction The gradient is:

[0039]

[0040] Subscript Representative Unit The eastern unit; substituting the above equation into equation (7) yields the unit surface. The discrete form of the diffusion flux is:

[0041]

[0042] Therefore, the coefficients are:

[0043]

[0044] in, Representation unit and unit Vectors between centroids Representation unit and unit The distance between the centroids;

[0045] Based on the above process processing unit surface , , By obtaining the coefficients and substituting them into equation (4), we obtain the algebraic form of the diffusion equation, namely:

[0046]

[0047] in:

[0048]

[0049] Simplifying the above equation, we get:

[0050]

[0051] in:

[0052]

[0053] Subscript Representative Unit Adjacent units ( , , , ), subscript Representative Unit unit surface ( , , , );

[0054] Step 3.1.2: Perform heat transfer analysis for different heat transfer boundary conditions;

[0055] Surrounding Unit Discretization yields the following relationship:

[0056]

[0057] The flux discretization of the internal unit surfaces has been completed, while the discretization of the boundary flux is for constructing information about the internal unit surfaces. A linear function, therefore:

[0058]

[0059] Subscript Represents the boundary element surface;

[0060] For heat transfer problems, there are Dirichlet, Neumann, and mixed boundary condition types. For a certain element surface of element C, consider the following three boundary conditions:

[0061] (1) For a certain element face of the element, for example, element face When using Dirichlet boundary conditions; when using Dirichlet boundary conditions, the boundary unknowns are given. The value of, that is:

[0062]

[0063] in this case:

[0064]

[0065] in:

[0066]

[0067] This represents the vector between the centroid of element C and the plane of the boundary element;

[0068] (2) For a certain element face of the element, such as the element face When using von Neumann boundary conditions; if the boundary conditions are given... If the flux or the normal gradient of the boundary element surface is given, then the boundary condition is called the Neumann boundary condition; under this condition, the given flux expression is:

[0069]

[0070] For the given boundary flux;

[0071] The above formula represents flux. The reason is:

[0072]

[0073] in:

[0074] (3) For a certain element face of the element, such as the element face When the boundary conditions are mixed;

[0075] If the given boundary conditions include the convection transport coefficient and surrounding reference values This condition is called the mixed boundary condition, that is:

[0076]

[0077] The above formula can be rewritten as:

[0078]

[0079] From this, we can obtain :

[0080]

[0081] Will Substituting the expression back into equation (23), the flux equation becomes:

[0082]

[0083] in:

[0084] Step 3.2: Construct an unstructured finite volume computational model for the steady-state heat transfer problem in a non-orthogonal unstructured mesh;

[0085] In general, unstructured meshes are non-orthogonal; therefore, surface vectors and the vector connecting the centroids of coplanar elements They are not collinear; in this case, the gradient along the interface normal cannot be written as... and The function, because it has a perpendicular to Components in direction;

[0086] In the case of orthogonal meshes, because and interface unit normal vector The gradients are collinear and perpendicular to the interface direction, and their expression is:

[0087] (28)

[0088] in , for , The vector between the two centroids and the origin. for and The distance between the two centroids;

[0089] For non-orthogonal meshes, including and The gradient direction of the expression must be along and The line connecting the two centroids;

[0090] if Indicates along and If the unit vector is the direction of the line connecting the two centroids, then:

[0091] (29)

[0092] in for and The vector between the two centroids;

[0093] therefore, The gradient in the direction is written as:

[0094] (30)

[0095] Therefore, in order to achieve flux linearization in non-orthogonal grids, the surface vector is written as two vectors. and The sum, that is:

[0096] (31)

[0097] in, and With the directions aligned, in order to be able to write part of the diffusion flux as and A function such that:

[0098] (32)

[0099] Decomposition using the minimum correction method ,make and Perpendicularity minimizes the non-orthogonal correction part in equation (32); as non-orthogonality increases, and The contribution to diffusion flux will decrease; The calculation formula is as follows:

[0100] (33)

[0101] in Represents a unit vector;

[0102] in for and The angle between the two sides; substituting their respective expressions into the semi-discrete equation of diffusion flux and expanding it, we obtain the final form of the discrete equation based on the unstructured grid:

[0103] (34)

[0104] in:

[0105] (35)

[0106] in For vectors The model;

[0107] The handling of boundary conditions for non-orthogonal meshes is similar to that for orthogonal meshes, but there are subtle differences, which are related to the presence of non-orthogonal diffusion. It also involves three types of boundary conditions:

[0108] (1) Dirichlet boundary conditions:

[0109] (36)

[0110] in:

[0111] (37)

[0112] (2) Neumann boundary conditions; The treatment of Neumann boundary conditions in non-orthogonal grids is the same as that in orthogonal grids, that is, the given boundary flux is directly added to the equation as a source term;

[0113] (3) Mixed boundary conditions:

[0114] (38)

[0115] in:

[0116] (39)

[0117] Step 3.3: Construct an unstructured finite volume computational model for the transient heat transfer problem;

[0118] For transient simulations, the governing equations need to be discretized in both space and time. Spatial discretization is performed in the spatial domain, just like the treatment of steady-state problems, while time discretization requires establishing time coordinates and calculating the derivatives or integrals of the transient terms based on these coordinates.

[0119] Typically, variables The governing equations for the transient behavior are:

[0120] (40)

[0121] Where the function This represents a space operator that includes all non-transient terms. Represents transient operators;

[0122] In unit Integrating equation (40) and discretizing it spatially, we get:

[0123] (41)

[0124] in Reference time The spatial discrete operator is written in the following algebraic form:

[0125] (42)

[0126] Using the backward Euler scheme, the derivative is derived using the Taylor series expansion. Represented as a function of discrete grid point values;

[0127] function exist The value at time t is written using the Taylor formula. Moment A function of the value and its derivative, i.e.:

[0128] (43)

[0129] After simplification, the expression for the first derivative is obtained:

[0130] (44)

[0131] In equation (44) Replace with Substituting this into the expression for the derivative in equation (41), the discrete equation becomes:

[0132] (45)

[0133] Using the algebraic relations of space operators, the complete algebraic form of the transient equation is:

[0134] ( ) + = (46)

[0135] in:

[0136]

[0137] (47)

[0138] superscript The superscript indicates the value of the variable at the previous time step. Indicates multiplication Unsteady-state coefficients;

[0139] Step 4: Using the unstructured finite volume heat transfer model established in Step 3, simulate the solidification heat transfer phenomenon of irregular continuously cast billets during continuous casting.

[0140] Step 4.1: Since there are many units, the required data storage space is large, so it is necessary to apply for storage space for the data in advance to prevent data overflow during subsequent calculations;

[0141] Step 4.2: Read the parameter information of the irregular continuous casting billet;

[0142] The system reads steel grade information, continuous casting machine structural parameters, and continuous casting process parameters to calculate the solidus temperature and liquidus temperature. The steel grade information includes the steel grade and chemical composition. The continuous casting machine structural parameters include the crystallizer height and secondary cooling zone structural parameters. The secondary cooling zone structural parameters include the number, length, inlet position, and outlet position of the secondary cooling zones. The continuous casting process parameters include the casting temperature, casting speed, crystallizer heat flux density, water flow rate and temperature in each secondary cooling zone, ambient temperature, and cross-sectional dimensions.

[0143] Step 4.3: Import the DAT format file exported in Step 2 into the unstructured finite volume heat transfer model and save it; the mesh information includes element information, node information, and element surface information; the element information includes the element number and the nodes contained in the element; the node information includes the node number and node coordinates; the element surface information includes the element surface number and the nodes that constitute the element surface;

[0144] Step 4.4: Calculate the relevant parameters of the mesh; the relevant parameters include element area, element centroid, element face length, and element face surface vector;

[0145] Step 4.4: Determine the time step; the time step is the time for each slice movement; the time step multiplied by the pulling speed is the distance the slice moves each time, and the cumulative movement distance is the position of the slice;

[0146] Step 4.5: Based on the mesh-related parameters obtained in Step 4.3, calculate the cell temperature gradient using the Green-Gaussian method or the gradient theorem;

[0147] Step 4.6: Determine the phase region of the unit based on the unit surface temperature and calculate the unit's physical property parameters; the phase region refers to the liquid phase region, solid phase region, and solid-liquid two-phase region; the physical property parameters include unit surface density, unit surface specific heat, unit surface solid fraction, and unit surface thermal conductivity.

[0148] Step 4.7: Calculate the coefficients according to equations (35) and (47) in Step 3. , , , , And substitute it into the discrete equation (46);

[0149] Step 4.8: Summarize the discrete equations in Step 4.7 and solve the discrete equation system using the Gauss-Seidel iterative method to obtain the unit surface temperature; after the solution is completed, check whether the slice position exceeds the air-cooled zone. If it has exceeded the air-cooled zone, the calculation ends and the heat transfer calculation of solidification of irregular continuous casting billet is completed. If it has not exceeded the air-cooled zone, the slice moves to the next position and proceeds to Step 4.5.

[0150] Step 5: Visualize and post-process the calculation results of solidification heat transfer of irregular continuous casting billets;

[0151] After the heat transfer calculation for solidification of irregular continuously cast billets is completed, the calculation results at different locations on the continuous casting machine are extracted and imported into Tecplot software for visualization to form a temperature cloud map. The temperature changes of feature points are extracted and imported into Origin software for visualization to form temperature change curves. The feature points include the center of the web surface, the center of the inner edge surface, the center of the narrow face film, and the center of the rounded corner surface for irregular billets; the center of the wide face surface, the center of the narrow face surface, the center of the billet, and the center of the rounded corner for round billets; and the center of the billet and the surface of the billet for round billets.

[0152] Beneficial technical effects of the present invention:

[0153] The basic idea of ​​the finite volume method is to divide the computational domain into a series of non-repeating control volumes, with each grid point surrounded by a control volume. Integrating the differential equation to be solved over each control volume yields a set of discrete equations, where the unknowns are the values ​​of the dependent variables at the grid points. This method is easy to understand and provides a direct physical interpretation. It can adapt to various complex regions, and the resulting matrix equations are diagonal, making them easier to solve and resulting in faster convergence and a solution rate far exceeding that of the finite element method.

[0154] This invention provides a finite volume calculation method for solidification heat transfer of irregular cross-section continuously cast billets. It fully considers the complex cross-sectional shape of irregular billets, employing general quadrilateral elements to better simulate the actual cross-section of irregular billets. It also considers the special case of square billets typically having rounded corners in actual production, using unstructured meshes. When the mesh is sufficiently dense, piecewise straight lines can be used to simulate arbitrary curves at the boundary of the calculation area, ensuring a certain level of calculation accuracy. The method uses a volumetric approach, which improves the calculation speed compared to the finite element method. Similarly, it fully considers the arc-shaped boundary of round billets, using unstructured meshes. When the mesh is sufficiently dense, piecewise straight lines can be used to simulate arbitrary curves at the boundary of the calculation area, ensuring a certain level of calculation accuracy. The method also uses a volumetric approach, which improves the calculation speed compared to the finite element method. Furthermore, by selecting appropriate boundary conditions based on different cold zones, it more accurately simulates the solidification heat transfer throughout the entire continuous casting production process of irregular continuously cast billets. Taking full account of the actual situation of continuous casting of irregular billets, the paper describes the temperature field changes during the continuous casting process of irregular billets, thus providing support for the uniform cooling of irregular billets and improving their quality. Attached Figure Description

[0155] Figure 1 A schematic flowchart of a finite volume calculation method for solidification heat transfer of an irregular cross-section continuous casting billet according to an embodiment of the present invention;

[0156] Figure 2. Schematic diagram of the cross-sectional dimensions of the irregular billet provided in the embodiment of the present invention;

[0157] Figure 3. Schematic diagram of 1 / 2 irregular billet grid division provided in an embodiment of the present invention;

[0158] Figure 4 shows the temperature distribution cloud map of the irregular billet at different positions on the continuous casting machine according to the embodiment of the present invention; wherein Figure (a) is the temperature distribution cloud map of the billet at the outlet of the crystallizer; Figure (b) is the temperature distribution cloud map of the billet at the outlet of the second cooling zone 1; Figure (c) is the temperature distribution cloud map of the billet at the outlet of the second cooling zone 2; Figure (d) is the temperature distribution cloud map of the billet at the outlet of the second cooling zone 3; Figure (e) is the temperature distribution cloud map of the billet at the outlet of the second cooling zone 4; and Figure (f) is the temperature distribution cloud map of the billet at the outlet of the second cooling zone 5.

[0159] Figure 5. Temperature variation curves of various feature points of the irregularly shaped billet provided in the embodiment of the present invention;

[0160] Figure 6. Comparison of calculated temperature and measured temperature at the center of the flange surface of the irregular billet provided in the embodiment of the present invention. Detailed Implementation

[0161] The present invention will be further described below with reference to the accompanying drawings and embodiments;

[0162] This embodiment takes the continuous casting process of a 450mm×350mm×90mm Q235 steel shaped billet in a domestic steel plant as an example. The finite volume calculation method for solidification heat transfer of irregular cross-section continuous casting billet of the present invention is used to calculate the temperature change at each moment during the continuous casting process of the Q235 steel shaped billet.

[0163] A non-structural finite volume method for simulating the solidification heat transfer of irregularly shaped billets in continuous casting, such as... Figure 1 As shown, the specific steps include:

[0164] Step 1: Construct a 1 / 2 irregular billet simulation calculation domain based on the cross-sectional dimensions of the irregular billet;

[0165] Based on the cross-sectional dimensions of the irregular billet, the coordinates of key points are determined and imported into ANSYS software. In ANSYS, the key points are connected to generate geometric surfaces, constructing a simulation calculation domain for the 1 / 2 irregular billet. The cross-sectional dimensions of the irregular billet include the flange thickness, wide face width, narrow face width, web length, web thickness, inner edge arc radius, flange radius, and fillet radius. The key points include the web endpoint, flange endpoint, inner edge arc endpoint, flange endpoint, and fillet endpoint.

[0166] In this embodiment, the width of the wide face of the irregular billet is 0.435m, the width of the narrow face is 0.320m, the flange thickness is 0.045m, the web length is 0.229m, the web thickness is 0.090m, the flange radius is 0.020m, the inner edge radius is 0.100m, and the corner radius is 0.010m. A schematic diagram of the cross-sectional dimensions of the irregular billet in this embodiment is shown below. Figure 2 As shown.

[0167] Step 2: Divide the unstructured mesh in the simulation computation domain and save the mesh information; the mesh information includes element information, node information, and element surface information;

[0168] Step 2.1: Select the element type and element shape in ANSYS, determine the element size, and then generate a non-mechanical mesh in the simulation domain;

[0169] In this embodiment, the element type is PLANE55, the element shape is quadrilateral, and the element size is 5mm. The region element meshing is automatically performed by ANSYS software. The schematic diagram of the 1 / 2 irregular blank mesh generation in this embodiment is shown below. Figure 3 As shown.

[0170] Step 2.2: Output the cell information and node information as a DAT format file and save it; the cell information includes the cell number and which nodes are contained in the cell; the node information includes the node number and node coordinates;

[0171] Step 2.3: Due to the complex cross-sectional shape of the irregular billet, ANSYS cannot use quadrilateral elements for meshing when performing region partitioning. There are some triangular elements. Therefore, it is necessary to determine whether all elements are quadrilateral elements. If not, the mesh needs to be re-generated and proceed to step 2.1.

[0172] Step 2.4: Extract and save the element surface information based on the element information and node information;

[0173] Extracting element surface information based on the relationship between elements, nodes, and faces, outputting it as a DAT format file and saving it; the relationship between elements, nodes, and faces is that an element consists of four nodes and has four element faces, and each element face consists of two nodes;

[0174] Step 3: To address the heat transfer problem during solidification in continuous casting of irregularly shaped billets, a non-structural finite volume heat transfer model is established.

[0175] Step 3.1: Discretize the diffusion equations in the regular Cartesian coordinate system and construct a finite volume calculation model for the steady-state heat transfer problem;

[0176] Step 3.1.1: Discrete steady-state heat transfer and diffusion equations;

[0177] Step 3.1.2: Perform heat transfer analysis for different heat transfer boundary conditions;

[0178] Analytical solutions to any ordinary differential equation or partial differential equation depend on constants determined by the boundary conditions. For heat transfer problems, there are Dirichlet, Neumann, and mixed boundary condition types. Boundary conditions are applied to boundary elements, which have one or more element faces on their boundaries.

[0179] Step 3.2: Construct an unstructured finite volume computational model for the steady-state heat transfer problem in a non-orthogonal unstructured mesh;

[0180] Step 3.3: Construct an unstructured finite volume computational model for the transient heat transfer problem;

[0181] Step 4: Using the unstructured finite volume heat transfer model established in Step 3, simulate the solidification heat transfer phenomenon of irregular billets during continuous casting.

[0182] Step 4.1: Read the steel grade information, continuous casting machine structural parameters, and continuous casting process parameters of the shaped billet, and calculate the solidus temperature and liquidus temperature. The steel grade information includes the steel grade and chemical composition. The continuous casting machine structural parameters include the crystallizer height and secondary cooling zone structural parameters. The secondary cooling zone structural parameters include the number, length, inlet position, and outlet position of the secondary cooling zones. The continuous casting process parameters include the casting temperature, casting speed, crystallizer heat flux density, water flow rate and temperature in each secondary cooling zone, ambient temperature, and shaped billet cross-sectional dimensions. The shaped billet cross-sectional dimensions include the flange thickness, wide face width, narrow face width, web length, web thickness, inner edge radius, flange radius, and corner radius.

[0183] In this embodiment, the steel grade is Q235, and the chemical composition of this steel is shown in Table 1.

[0184] chemical composition C Si Mn P S Quality score (%) 0.19 0.21 0.47 0.025 0.027

[0185] In this embodiment, the height of the mold of the shaped billet continuous casting machine is 0.7m; there are 5 secondary cooling zones, of which the length of secondary cooling zone 1 is 0.647m, and the inlet and outlet are 0.7m and 1.347m from the meniscus, respectively; the length of zone 2 is 1.486m, and the inlet and outlet are 1.347m and 2.833m from the meniscus, respectively; the length of zone 3 is 2.337m, and the inlet and outlet are 2.833m and 5.170m from the meniscus, respectively; the length of zone 4 is 2.378m, and the inlet and outlet are 5.170m and 7.548m from the meniscus, respectively; and the length of zone 5 is 2.224m, and the inlet and outlet are 7.548m and 9.772m from the meniscus, respectively. The casting temperature of the Q235 shaped billet was 1545℃, and the casting speed was 0.9 m / min. The heat flux density on the wide face of the crystallizer was 247 W / m², and the heat flux density on the narrow face was 243.677 W / m². The cooling conditions in the secondary cooling zone are shown in Table 2. The secondary cooling water temperature was 25℃, and the ambient temperature in the air cooling zone was 90℃.

[0186] Second cooling zone 1 2 3 4 5 Inner arc water flow rate (L / min) 36 33 21 13 13 External arc water flow rate (L / min) 36 33 21 13 13 Side arc water flow rate (L / min) 40 37 19 8.4 8.4 Total water volume (L / min) 112 103 60 34..4 34.4

[0187] In this embodiment, the calculation formula for the corresponding solid-liquid phase temperature is selected based on the C content of Q235 steel:

[0188] (48)

[0189] (49)

[0190] in Ts is the liquidus temperature; [C], [Si], [Mn], [P], [S], [Cr], [Al], [Ni], [V], and [Mo] are the mass fractions of the corresponding elements.

[0191] Step 4.2: Import the mesh information of the 2D geometric model of the 1 / 2 irregular billet;

[0192] Import the DAT format file exported in step 2 into the heat transfer model and save it; the mesh information includes element information, node information, and element surface information; the element information includes the element number and the nodes contained in the element; the node information includes the node number and the node coordinates; the element surface information includes the element surface number and the nodes that make up the element surface;

[0193] Step 4.3: Calculate the relevant parameters of the mesh; the relevant parameters include element area, element centroid, element face length, and element face vector;

[0194] Step 4.4: Determine the time step; the time step is the time it takes for the slice (called a slice because the two-dimensional model has no thickness) to move each time; the time step multiplied by the pulling speed is the distance the slice moves each time, and the cumulative distance moved is the position of the slice;

[0195] In this embodiment, the time step is 0.1s and the pulling speed is 0.9m / min.

[0196] Step 4.5: Based on the mesh-related parameters obtained in Step 4.3, calculate the cell temperature gradient using the Green-Gaussian method or the gradient theorem;

[0197] Step 4.6: Determine the phase region of the unit based on the unit surface temperature and calculate the unit's physical property parameters; the phase region refers to the liquid phase region, solid phase region, and solid-liquid two-phase region; the physical property parameters include unit surface density, unit surface specific heat, unit surface solid fraction, and unit surface thermal conductivity.

[0198] In this embodiment, as shown in step 4.2, the solidus temperature is 1485℃ and the liquidus temperature is 1515℃. The phase region of the unit surface is determined based on the unit surface temperature, and the physical property parameters of the unit are calculated using appropriate formulas.

[0199] Step 4.7: Calculate the coefficients according to equations (35) and (47) in Step 3. , , , , And substitute it into the discrete equation (46);

[0200] Step 4.8: Summarize the discrete equations in Step 4.7 and solve the discrete equation system using the Gauss-Seidel iterative method to obtain the unit surface temperature; after the solution is completed, determine whether the slice position exceeds the air-cooled zone. If it has exceeded the air-cooled zone, the calculation ends and the solidification heat transfer calculation of the special-shaped billet is completed. If it has not exceeded the air-cooled zone, the slice moves to the next position and proceeds to Step 4.5.

[0201] This embodiment uses the Gauss-Seidel iterative method to solve the discrete equation system. The error is the square root of the sum of the squares of the differences between the two iterations of all nodes, with an error tolerance of 0.001. To improve computational efficiency, the parallel_for function in OpenCV is used for CPU parallel computation.

[0202] Step 5: Visualize and post-process the calculation results of solidification heat transfer in continuous casting of irregular billets;

[0203] After the solidification heat transfer calculation of the irregular billet is completed, the calculation results at different locations of the continuous casting machine are extracted and imported into Tecplot software for visualization to form a temperature cloud map; the temperature changes at the center of the web, the center of the inner edge, the center of the narrow face, and the center of the rounded corner are extracted and imported into Origin software for visualization to form a temperature change curve.

[0204] This embodiment provides temperature distribution cloud maps of 450mm×350mm×90mm Q235 special-shaped billets at different locations on a continuous casting machine from a domestic steel plant, such as... Figure 4 As shown, Figure (a) is the temperature distribution cloud map of the slab at the outlet of the crystallizer; Figure (b) is the temperature distribution cloud map of the slab at the outlet of the second cooling zone 1; Figure (c) is the temperature distribution cloud map of the slab at the outlet of the second cooling zone 2; Figure (d) is the temperature distribution cloud map of the slab at the outlet of the second cooling zone 3; Figure (e) is the temperature distribution cloud map of the slab at the outlet of the second cooling zone 4; and Figure (f) is the temperature distribution cloud map of the slab at the outlet of the second cooling zone 5. The corresponding temperature change curves for each characteristic point are shown below. Figure 5 As shown. Comparison of calculated and measured temperatures at the center of the flange surface, as shown. Figure 6 As shown. From Figure 6 It can be seen that the simulation results are basically consistent with the actual temperatures. Due to the shape and cooling characteristics of the irregular billet, as cooling proceeds, the high-temperature region at the center of the billet gradually decreases and moves towards the inner edge. Because the inner edge is concave, heat transfer is poor there, and the surface temperature is higher than at other locations on the surface. The corners of the irregular billet experience rapid heat loss due to two-dimensional heat transfer, resulting in the lowest corner temperatures.

Claims

1. A finite volume calculation method for solidification heat transfer of irregular cross-section continuously cast billets, characterized in that, Specifically, the following steps are included: Step 1: Construct a 1 / 2 continuous casting billet simulation calculation domain based on the cross-sectional dimensions of the irregular cross-section continuous casting billet; the irregular cross-section continuous casting billet includes shaped billets, rounded square billets, and round billets; Step 2: Divide the unstructured mesh in the 1 / 2 continuous casting billet simulation calculation domain established in Step 1 and save the mesh information; the mesh information includes element information, node information, and element surface information; Step 3: To address the heat transfer problem during the solidification of continuously cast billets, a non-structural finite volume heat transfer model is established. Step 4: Using the unstructured finite volume heat transfer model established in Step 3, simulate the solidification heat transfer phenomenon of irregular continuously cast billets during continuous casting. Step 5: Visualize and post-process the calculation results of solidification heat transfer of irregular continuous casting billets; Step 3 specifically involves: Step 3.1: Discretize the diffusion equations in the regular Cartesian coordinate system and construct a finite volume calculation model for the steady-state heat transfer problem; Step 3.2: Construct an unstructured finite volume computational model for the steady-state heat transfer problem in a non-orthogonal unstructured mesh; In general, unstructured meshes are non-orthogonal; therefore, surface vectors and the vector connecting the centroids of coplanar elements They are not collinear; in this case, the gradient along the interface normal cannot be written as... and The function, because it has a perpendicular to Components in direction; In the case of orthogonal meshes, because and interface unit normal vector The gradients are collinear and perpendicular to the interface direction, and their expression is: （28） in , for , The vector between the two centroids and the origin. for and The distance between the two centroids; For non-orthogonal meshes, including and The gradient direction of the expression must be along and The line connecting the two centroids; if Indicates along and If the unit vector is the direction of the line connecting the two centroids, then: （29） in for and The vector between the two centroids; therefore, The gradient in the direction is written as: （30） Therefore, in order to achieve flux linearization in non-orthogonal grids, the surface vector is written as two vectors. and The sum, that is: （31） in, and With the directions aligned, in order to be able to write part of the diffusion flux as and A function such that: （32） Decomposition using the minimum correction method ,make and Perpendicularity minimizes the non-orthogonal correction part in equation (32); as non-orthogonality increases, and The contribution to diffusion flux will decrease; The calculation formula is as follows: （33） in Represents a unit vector; in for and The angle between the two sides; substituting their respective expressions into the semi-discrete equation of diffusion flux and expanding it, we obtain the final form of the discrete equation based on the unstructured grid: （34） in: （35） in For vectors The model; The handling of boundary conditions for non-orthogonal meshes is similar to that for orthogonal meshes, but there are subtle differences, which are related to the presence of non-orthogonal diffusion. It also involves three types of boundary conditions: (1) Dirichlet boundary conditions: （36） in: （37） (2) Neumann boundary conditions; The treatment of Neumann boundary conditions in non-orthogonal grids is the same as that in orthogonal grids, that is, the given boundary flux is directly added to the equation as a source term; (3) Mixed boundary conditions: （38） in: （39） Step 3.3: Construct an unstructured finite volume computational model for the transient heat transfer problem; For transient simulations, the governing equations need to be discretized in both space and time. Spatial discretization is performed in the spatial domain, just like the treatment of steady-state problems, while time discretization requires establishing time coordinates and calculating the derivatives or integrals of the transient terms based on these coordinates. Typically, variables The governing equations for the transient behavior are: （40） Where the function This represents a space operator that includes all non-transient terms. Represents transient operators; In unit Integrating equation (40) and discretizing it spatially, we get: （41） in Reference time The spatial discrete operator is written in the following algebraic form: （42） Using the backward Euler scheme, the derivative is derived using the Taylor series expansion. Represented as a function of discrete grid point values; function exist The value at time t is written using the Taylor formula. Moment A function of the value and its derivative, i.e.: （43） After simplification, the expression for the first derivative is obtained: （44） In equation (44) Replace with Substituting this into the expression for the derivative in equation (41), the discrete equation becomes: （45） Using the algebraic relations of space operators, the complete algebraic form of the transient equation is: （46） in: （47） superscript The superscript indicates the value of the variable at the previous time step. Indicates multiplication The coefficients of the unsteady-state term.

2. The finite volume calculation method for solidification heat transfer of irregular cross-section continuously cast billets according to claim 1, characterized in that, The construction of the simulation calculation domain for the 1 / 2 continuous casting billet in step 1 is specifically as follows: Based on the cross-sectional dimensions, the coordinates of key points are determined and imported into ANSYS software. In ANSYS, the key points are connected to generate geometric surfaces, constructing a 1 / 2 simulation computational domain. The cross-sectional dimensions for irregularly shaped billets include flange thickness, wide face width, narrow face width, web length, web thickness, inner edge radius, flange radius, and fillet radius; for rounded square billets, they include wide face width, narrow face width, and fillet radius; and for round billets, they include the round billet radius. The key points for irregularly shaped billets include web endpoints, flange endpoints, inner edge radius endpoints, flange endpoints, and fillet endpoints; and for rounded square billets, they include fillet endpoints, wide face endpoints, and narrow face endpoints. For round blanks, including the center.

3. The finite volume calculation method for solidification heat transfer of irregular cross-section continuously cast billets according to claim 1, characterized in that, Step 2 is as follows: Step 2.1: Select the element type and element shape in ANSYS, and determine the element size. Then, generate a non-mechanical mesh in the continuous casting billet simulation calculation domain. Step 2.2: Output the cell information and node information as a DAT format file and save it; the cell information includes the cell number and which nodes are contained in the cell; the node information includes the node number and node coordinates; Step 2.3: Extract and save the element surface information based on the element information and node information; Because the cross-sectional shape of the irregular continuous casting billet is relatively complex, ANSYS cannot use quadrilateral elements for meshing when performing region partitioning. There are some triangular elements. Therefore, it is necessary to determine whether all the elements are quadrilateral elements. If not, the mesh needs to be re-partitioned and proceed to step 2.

1. Extract the element surface information by the relationship between elements, nodes and surfaces, output it as a DAT format file and save it; the relationship between elements, nodes and surfaces is that an element consists of four nodes and has four element surfaces, and each element surface consists of two nodes.

4. The finite volume calculation method for solidification heat transfer of irregular cross-section continuously cast billets according to claim 1, characterized in that, Step 3.1 specifically involves: Step 3.1.1: Discrete steady-state heat transfer and diffusion equations; The fundamental variable in the heat transfer process is temperature, which is a function of the geometric position within the object and time. First, consider a regular Cartesian mesh over a simple rectangular domain. Based on this mesh, the steady-state diffusion equation is: ； in It represents a scalar quantity, which is temperature in heat transfer problems; It is the diffusion coefficient, which is the thermal conductivity coefficient in heat transfer problems, W / (m·K); Represents the volume per unit volume within this computational domain The generation rate, i.e., the source term, is the internal heat source in heat transfer problems, W / The equation can be written in its general form as follows: ； in It is the diffusion flux; Taking unit C as an example, equation (1) is applied to unit C. Discretize it and transform it into: ； Subscript Representative Unit Each unit surface on, Represents the surface vector. Representative unit C The generation rate, i.e., the source term, Representative Unit Controlling volume, Expanding the above equation, we get: ； Subscript , , , The four faces of element C represent the cardinal directions (north, south, east, and west). For a uniform Cartesian mesh, the surface vector of an element face is calculated using the following formula: ； in represent Increment in the axial direction, represent Increment in the axial direction, represent Unit vector along the axis, represent Unit vector along the axis; Therefore, as an example, the diffusion flux of the eastern unit surface is: ； In the discretization of the conservation equation, if a single integration point approach is used, the discrete form of the diffusion flux is: ； Assumption If the element C exhibits a linear variation between its centroids, then the element surface... upper edge direction The gradient is: ； Subscript Representative Unit The eastern unit; substituting the above equation into equation (7) yields the unit surface. The discrete form of the diffusion flux is: ； Therefore, the coefficients are: ； in, Representation unit and unit Vectors between centroids Representation unit and unit The distance between the centroids; Based on the above process processing unit surface , , By obtaining the coefficients and substituting them into equation (4), we obtain the algebraic form of the diffusion equation, namely: ； in: ； Simplifying the above equation, we get: ； in: ； Subscript Representative Unit Adjacent units ( , , , ), subscript Representative Unit unit surface ( , , , ); Step 3.1.2: Perform heat transfer analysis for different heat transfer boundary conditions; Surrounding Unit Discretization yields the following relationship: ； The flux discretization of the internal unit surfaces has been completed, while the discretization of the boundary flux is for constructing information about the internal unit surfaces. A linear function, therefore: ； Subscript Represents the boundary element surface; For heat transfer problems, there are Dirichlet, Neumann, and mixed boundary condition types. For a certain element surface of element C, consider the following three boundary conditions: (1) When a certain element surface of the element is subject to Dirichlet boundary conditions; Dirichlet boundary conditions provide unknowns at the boundary. The value of, that is: ； in this case: ； in: ; This represents the vector between the centroid of element C and the plane of the boundary element; (2) When a certain element surface of the element is subject to Neumann boundary conditions; if the boundary conditions are given... If the flux or the normal gradient of the boundary element surface is given, then the boundary condition is called the Neumann boundary condition; under this condition, the given flux expression is: ； For the given boundary flux; The above formula represents flux. The reason is: ； in: ； (3) When a certain element surface of the element is subject to mixed boundary conditions; If the given boundary conditions include the convection transport coefficient and surrounding reference values This condition is called the mixed boundary condition, that is: ； The above formula can be rewritten as: ； From this, we can obtain : ； Will Substituting the expression back into equation (23), the flux equation becomes: ； in: 。 5. The finite volume calculation method for solidification heat transfer of irregular cross-section continuously cast billets according to claim 1, characterized in that, Step 4 is as follows: Step 4.1: Since there are many units, the required data storage space is large, so it is necessary to apply for storage space for the data in advance to prevent data overflow during subsequent calculations; Step 4.2: Read the parameter information of the irregular continuous casting billet; The system reads steel grade information, continuous casting machine structural parameters, and continuous casting process parameters to calculate the solidus temperature and liquidus temperature. The steel grade information includes the steel grade and chemical composition. The continuous casting machine structural parameters include the crystallizer height and secondary cooling zone structural parameters. The secondary cooling zone structural parameters include the number, length, inlet position, and outlet position of the secondary cooling zones. The continuous casting process parameters include the casting temperature, casting speed, crystallizer heat flux density, water flow rate and temperature in each secondary cooling zone, ambient temperature, and cross-sectional dimensions. Step 4.3: Import the DAT format file exported in Step 2 into the unstructured finite volume heat transfer model and save it; the mesh information includes element information, node information, and element surface information; the element information includes the element number and the nodes contained in the element; the node information includes the node number and node coordinates; the element surface information includes the element surface number and the nodes that constitute the element surface; Step 4.4: Calculate the relevant parameters of the mesh; the relevant parameters include element area, element centroid, element face length, and element face surface vector; Step 4.4: Determine the time step; the time step is the time for each slice movement; the time step multiplied by the pulling speed is the distance the slice moves each time, and the cumulative movement distance is the position of the slice; Step 4.5: Based on the mesh-related parameters obtained in Step 4.3, calculate the cell temperature gradient using the Green-Gaussian method or the gradient theorem; Step 4.6: Determine the phase region of the unit based on the unit surface temperature and calculate the unit's physical property parameters; the phase region refers to the liquid phase region, solid phase region, and solid-liquid two-phase region; the physical property parameters include unit surface density, unit surface specific heat, unit surface solid fraction, and unit surface thermal conductivity. Step 4.7: Calculate the coefficients according to equations (35) and (47) in Step 3. , , , , And substitute it into the discrete equation (46); Step 4.8: Summarize the discrete equations in Step 4.7 and solve the discrete equation system using the Gauss-Seidel iterative method to obtain the unit surface temperature; after the solution is completed, check whether the slice position exceeds the air-cooled zone. If it has exceeded the air-cooled zone, the calculation ends and the heat transfer calculation of solidification of irregular continuous casting billet is completed. If it has not exceeded the air-cooled zone, the slice moves to the next position and proceeds to Step 4.

5.

6. The finite volume calculation method for solidification heat transfer of irregular cross-section continuously cast billets according to claim 1, characterized in that, Step 5 describes the visualization and post-processing of the calculation results for the solidification heat transfer of irregular continuously cast billets: After the heat transfer calculation for solidification of irregular continuously cast billets is completed, the calculation results at different locations on the continuous casting machine are extracted and imported into Tecplot software for visualization to form a temperature cloud map. The temperature changes of feature points are extracted and imported into Origin software for visualization to form temperature change curves. The feature points include the center of the web surface, the center of the inner edge surface, the center of the narrow face film, and the center of the rounded corner surface for irregular billets; the center of the wide face surface, the center of the narrow face surface, the center of the billet, and the center of the rounded corner for round billets; and the center of the billet and the surface of the billet for round billets.

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