Method and system for calculating temperature distribution and solid phase rate distribution of variable cross-section continuous casting billet

Abstract

The application provides a variable cross-section continuous casting billet temperature distribution and solid phase rate distribution calculation method and system, wherein the method comprises the following steps: constructing an initial three-dimensional geometric model of a continuous casting billet in a variable cross-section crystallizer through a preset model construction tool; performing grid division pretreatment on the initial three-dimensional geometric model to form a grid format variable cross-section continuous casting billet model; determining the velocity distribution of fluid in the grid format variable cross-section continuous casting billet model based on a preset fluid mechanics analysis tool; and determining the temperature field distribution and solid phase rate distribution of the continuous casting billet in the variable cross-section crystallizer through the fluid mechanics analysis tool based on the velocity distribution of the fluid in the grid format variable cross-section continuous casting billet model. The variable cross-section continuous casting billet temperature distribution and solid phase rate distribution calculation method and system provided by the application can solve the problems of low calculation precision and time-consuming of the existing variable cross-section continuous casting billet heat transfer and solidification calculation method.

Description

Technical Field

[0001] This invention relates to the field of iron and steel metallurgy technology, and more specifically, to a method and system for calculating the temperature distribution and solid fraction distribution of a variable cross-section continuous casting billet. Background Technology

[0002] Thin slab continuous casting and rolling is a near-net-shape, high-efficiency steel production technology. Variable cross-section slab molds (such as funnel-shaped slab molds) are the core equipment for thin slab continuous casting. The upper part of the funnel-shaped mold has a sufficient cross-sectional area to accommodate the insertion of the immersion nozzle and the melting of the protective slag, while the lower cross-section is close to the dimensions of the thin slab. The molten steel flows turbulently within the funnel-shaped mold, and the slab shell formed within it is constrained by the casting and mold, causing deformation and varying velocities at different points within the shell. Due to the different cross-sectional areas of the funnel-shaped mold, the throughput at each section also varies. In heat transfer analysis, the complex geometry and uneven velocity distribution of the funnel-shaped mold make simulating the solidification process difficult.

[0003] Light and heavy pressing techniques are important methods for improving the internal quality of large-section billets and slabs, and are widely used in steel production. However, the effects of large deformations and significant changes in cross-sectional dimensions on heat transfer and solidification are not considered in the commonly used offline process design or online control calculation of the billet temperature field. The calculation mesh remains unchanged, which may cause calculation errors.

[0004] In the simulation of the solidification process of conventional continuously cast billets, the temperature gradient along the drawing direction (Z-axis) is very small, so the thermal conductivity in this direction can be ignored, and only the thermal conductivity in the other two directions (X and Y) of the billet cross-section is considered. Assuming the cross-sectional area of ​​the billet remains constant, the velocity of each point along the Z-axis is the same, while the velocities in the X and Y directions are zero, the thin-plate movement method can be used to simplify the three-dimensional heat transfer problem of billet solidification into a two-dimensional thermal conductivity analysis. For slabs, billets, etc., those skilled in the art have developed relevant billet temperature field calculation programs to formulate secondary cooling processes, and based on these, dynamic water distribution and dynamic light pressure reduction models have been developed for computer process control, achieving good results.

[0005] However, if the billet undergoes heavy pressure and its dimensions change significantly, the above assumptions will deviate considerably from reality, affecting the accuracy of the calculations. Under heavy pressure, the cross-section of the billet continuously decreases, the material's velocity along the Z-axis increases, and it also moves in the X and Y directions, altering the velocity distribution within the material and causing changes in heat transfer phenomena. To reduce the deviation between the governing equations describing the physical process and reality, a three-dimensional heat transfer equation must be used to analyze the solidification of continuously cast billets with variable cross-sections. Existing three-dimensional heat transfer analyses in funnel-shaped molds generally employ commercial fluid dynamics software, such as Fluent, Flow3D, Fidap, and Ansys. These calculations consider turbulent flow of molten steel and even electromagnetic effects during casting, resulting in numerous governing equations, a huge computational workload, and high computational costs. Typically, the number of computational grids ranges from hundreds of thousands to millions, and the time step to satisfy the convergence and stability conditions is very small, approximately 1.0. -4 Even on multi-core numerical simulation workstations, a single calculation can take anywhere from several seconds to tens of days to complete. In-depth research by metallurgists has revealed the flow and heat transfer laws in the continuous casting process, contributing to the development of production processes and the improvement of product quality. Regarding the problem of continuous casting billets under heavy pressure, most studies have also used commercial software for thermo-mechanical coupling calculations. However, the software costs, hardware costs, and time costs are considerable, making it unsuitable for the engineering requirements and frequent process design needs.

[0006] Therefore, there is an urgent need for a method that can achieve three-dimensional heat transfer analysis in variable cross-section crystallizers while also saving costs and time to a certain extent. Summary of the Invention

[0007] In view of the above problems, the purpose of this invention is to provide a method and system for calculating the temperature distribution and solid fraction distribution of variable cross-section continuous casting billets, so as to solve the problems of low calculation accuracy and time-consuming calculation of heat transfer and solidification of existing variable cross-section continuous casting billets.

[0008] According to one aspect of the present invention, a method for calculating the temperature distribution and solids fraction distribution of a variable cross-section continuously cast billet is provided, comprising:

[0009] An initial three-dimensional geometric model of the continuously cast billet in the variable cross-section crystallizer is constructed using a preset model building tool.

[0010] The initial three-dimensional geometric model is preprocessed by meshing to form a mesh-variable cross-section continuous casting billet model;

[0011] The velocity distribution of the fluid in the grid-type variable cross-section continuous casting billet model is determined based on a preset fluid dynamics analysis tool.

[0012] Based on the velocity distribution of the fluid in the grid-type variable cross-section continuous casting billet model, the temperature field distribution and solid fraction distribution of the continuous casting billet in the variable cross-section crystallizer are determined by the fluid dynamics analysis tool.

[0013] Furthermore, a preferred embodiment is that the preset model building tool is FreeCAD; and the process of building the initial three-dimensional geometric model of the continuous casting billet in the variable cross-section crystallizer using FreeCAD includes:

[0014] The outer contour curve and the straight lines around the continuous casting billet on the symmetry plane in the variable cross-section crystallizer are drawn using FreeCAD, and NURBS surfaces are generated using the surface generation tool in FreeCAD.

[0015] The initial three-dimensional geometric model is constructed based on the outer contour curve and surrounding straight lines on the symmetry plane of the continuously cast billet in the variable cross-section crystallizer, as well as the NURBS surface.

[0016] Furthermore, in a preferred embodiment, the process of performing mesh preprocessing on the initial three-dimensional geometric model to form a mesh-variable cross-section continuous casting billet model includes:

[0017] Save the initial three-dimensional geometric model in Step format;

[0018] The initial three-dimensional geometric model of the Step format is processed by Gmsh to form the Gmsh grid-type variable cross-section continuous casting billet model.

[0019] Furthermore, in a preferred embodiment, the process of performing mesh preprocessing on the initial three-dimensional geometric model to form a mesh-variable cross-section continuous casting billet model further includes:

[0020] The Gmsh grid-type variable cross-section continuous casting billet model was converted into an OpenFOAM grid-type variable cross-section continuous casting billet model using the GmshToFoam tool.

[0021] Furthermore, a preferred embodiment is that the preset fluid dynamics analysis tool is OpenFOAM; and the process of determining the velocity distribution of the fluid in the grid-variable cross-section continuous casting billet model based on the preset fluid dynamics analysis tool includes:

[0022] The velocity distribution of the fluid in the OpenFOAM grid-variable cross-section continuous casting billet model is calculated using the incompressible fluid calculation program icoFOAM in OpenFOAM.

[0023] Furthermore, in a preferred embodiment, determining the temperature field distribution and solid fraction distribution of the continuous casting billet in the variable cross-section mold using the fluid dynamics analysis tool based on the velocity distribution of the fluid in the grid-type variable cross-section continuous casting billet model includes:

[0024] Based on the fluid velocity distribution in the grid-type variable cross-section continuous casting billet model, the temperature field distribution and the solid fraction distribution are calculated using the scalarTransportFoam solver in OpenFOAM. During the calculation, the solidification heat transfer control equation for the grid-type variable cross-section continuous casting billet model is:

[0025]

[0026] In the above formula, κ eff It is the equivalent thermal conductivity, where T is temperature, ρ is density, L is latent heat of solidification, and g is the density. l C is the liquid phase fraction. p For specific heat, The heat is generated due to the movement of the billet and changes in the liquid phase fraction. This refers to the release of latent heat caused by the change in liquid phase fraction over time.

[0027] Furthermore, in a preferred embodiment, the process of calculating the temperature field distribution and the solid fraction distribution using the OpenFOAM scalarTransportFoam solver further includes:

[0028] A time derivative term is added to the solidification heat transfer control equation, and the source term is linearized. The calculation formula for the source term linearization is as follows: in, T s and T l These are the solidus and liquidus temperatures, T 0 Δt and Δt represent the temperature at the previous moment and the calculation time step, respectively.

[0029] On the other hand, the present invention also provides a system for calculating the temperature distribution and solids content distribution of a variable cross-section continuously cast billet, comprising:

[0030] The modeling unit is used to construct the initial three-dimensional geometric model of the continuously cast billet in the variable cross-section crystallizer using a preset model building tool.

[0031] The preprocessing unit is used to perform mesh preprocessing on the initial three-dimensional geometric model to form a mesh-variable cross-section continuous casting billet model;

[0032] The velocity distribution calculation unit is used to determine the velocity distribution of the fluid in the grid-variable cross-section continuous casting billet model based on a preset fluid dynamics analysis tool.

[0033] The final calculation unit is used to determine the temperature field distribution and solid fraction distribution of the continuous casting billet in the variable cross-section mold based on the velocity distribution of the fluid in the grid-type variable cross-section continuous casting billet model, using the fluid dynamics analysis tool.

[0034] Furthermore, in a preferred embodiment, the preprocessing unit includes:

[0035] The first processing unit is used to save the initial three-dimensional geometric model in Step format and process the initial three-dimensional geometric model in Step format using Gmsh to form a Gmsh grid-type variable cross-section continuous casting billet model.

[0036] The second processing unit is used to convert the Gmsh grid-type variable cross-section continuous casting billet model into the OpenFOAM grid-type variable cross-section continuous casting billet model using the GmshToFoam tool.

[0037] Furthermore, in a preferred embodiment, the final calculation unit is used to calculate the temperature field distribution and solid fraction distribution of the continuously cast billet in the variable cross-section crystallizer using the solidification heat transfer control equation; and the final calculation unit further includes:

[0038] An inertial term addition unit is used to add inertial terms to the solidification heat transfer control equation:

[0039] The source term linearization processing unit is used to perform source term linearization processing on the solidification heat transfer control equation.

[0040] The method and system for calculating the temperature distribution and solids fraction distribution of continuously cast slabs according to the present invention establish a simplified calculation method for the solidification of continuously cast slabs in a variable cross-section crystallizer using open-source software FreeCAD, Gmsh, and OpenFOAM, providing an economical calculation tool for secondary cooling calculations in thin slab continuous casting. Furthermore, based on scalarTransportFoam, a three-dimensional heat transfer and solidification calculation program was developed, and a calculation method to accelerate convergence was proposed for three-dimensional nonlinear heat transfer equations involving the release of latent heat of solidification. In addition, the existing OpenFOAM standard program can only be used to solve differential equations with constant coefficients and lacks processing for variable physical property parameters. This invention addresses the temperature-dependent changes in physical property parameters and non-uniform heat transfer boundaries, meeting the simulation requirements for the solidification process of continuously cast slabs. Finally, by using the method and system for calculating the temperature distribution and solids fraction distribution of continuously cast slabs provided by this invention to calculate the temperature field of continuously cast slabs with variable cross-sections, the calculation efficiency and accuracy can be significantly improved.

[0041] To achieve the foregoing and related objectives, one or more aspects of the invention include the features that will be described in detail below and particularly pointed out in the claims. 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 include all such aspects and their equivalents. Attached Figure Description

[0042] 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 and the contents of the claims, and with a more complete understanding of the invention. In the drawings:

[0043] Figure 1 This is a flowchart illustrating the method for calculating the temperature distribution and solid fraction distribution of a variable cross-section continuously cast billet according to an embodiment of the present invention.

[0044] Figure 2 A contour diagram of a continuously cast billet in a funnel-shaped crystallizer of 1 / 4 of the present invention;

[0045] Figure 3 A model diagram of a continuously cast billet with a grid-type variable cross-section, representing a 1 / 4-inch funnel-shaped crystallizer according to an embodiment of the present invention;

[0046] Figure 4 This is a graph showing the velocity variation along the center line of the continuously cast billet in a funnel-shaped crystallizer according to an embodiment of the present invention.

[0047] Figure 5 This is a temperature distribution diagram on the cross-section of the continuously cast billet at the outlet of the funnel-shaped crystallizer according to an embodiment of the present invention;

[0048] Figure 6 The liquid phase distribution on the cross-section of the continuously cast billet at the outlet of the funnel-shaped crystallizer according to an embodiment of the present invention.

[0049] In all the accompanying drawings, the same reference numerals indicate similar or corresponding features or functions. Detailed Implementation

[0050] The following is a detailed description of the method and system for calculating the temperature distribution and solid fraction distribution of the variable cross-section continuous casting billet provided by the present invention.

[0051] Specifically, Figure 1 The flowchart illustrates the method for calculating the temperature distribution and solids fraction distribution of a variable cross-section continuously cast billet according to an embodiment of the present invention. Figure 2 The outline of a continuously cast billet in a funnel-shaped crystallizer according to an embodiment of the present invention is shown. Figure 3 A 1 / 4-inch funnel-shaped crystallizer grid-type variable cross-section continuous casting billet model according to an embodiment of the present invention is shown. Figure 4The curve showing the velocity variation along the center line of the continuously cast billet in a funnel-shaped crystallizer according to an embodiment of the present invention is shown. Figure 5 The degree distribution on the cross-section of the continuously cast billet at the outlet of the funnel-shaped crystallizer according to an embodiment of the present invention is shown. Figure 6 The liquid phase distribution on the cross-section of the continuously cast billet at the outlet of the funnel-shaped crystallizer according to an embodiment of the present invention is shown.

[0052] As shown in the figure Figures 1 to 6 As shown in the figure, the method for calculating the temperature distribution and solids fraction distribution of a variable cross-section continuously cast billet provided by the present invention includes:

[0053] S110: Construct the initial three-dimensional geometric model of the continuously cast billet in the variable cross-section crystallizer using the preset model construction tool.

[0054] It should be noted that in the actual modeling process, since the crystallizer has a variable cross-section structure (such as a funnel-shaped crystallizer), in order to accurately construct the three-dimensional geometric model of the continuous casting billet in the variable cross-section crystallizer, the default model building tool can be FreeCAD. FreeCAD is a free and open-source software used for 2D and 3D modeling. This software supports multiple formats, such as IGES, STEP, OBJ, DXF, SVG, STL and DAE. Of course, other tools can also be used, as long as they can achieve the construction of the three-dimensional geometric model of the continuous casting billet in the variable cross-section crystallizer. This will not be elaborated on here.

[0055] Specifically, the preset model building tool is FreeCAD; and the process of building the initial three-dimensional geometric model of the continuous casting billet in the variable cross-section crystallizer using FreeCAD includes:

[0056] The outer contour curve and the straight lines around the continuous casting billet on the symmetry plane in the variable cross-section crystallizer are drawn using FreeCAD, and NURBS surfaces are generated using the surface generation tool in FreeCAD.

[0057] The initial three-dimensional geometric model is constructed based on the outer contour curve and surrounding straight lines on the symmetry plane of the continuously cast billet in the variable cross-section crystallizer, as well as the NURBS surface.

[0058] More specifically, taking a funnel-shaped crystallizer as an example, the surface of the funnel-shaped crystallizer is very complex and it is difficult to describe its surface shape with simple mathematical formulas. It is also not easy to generate a mesh for numerical calculation. Furthermore, the built-in geometric modeling and mesh generation tools of OpenFOAM, a preset fluid dynamics analysis tool used in later calculations, are difficult to use for geometric entities with complex surfaces. Therefore, this invention uses FreeCAD to construct the initial three-dimensional geometric model of the continuous casting billet in the funnel-shaped crystallizer.

[0059] Furthermore, since OpenFOAM provides rich interfaces, the geometric model of the continuous casting billet in the funnel-shaped crystallizer can be completed first using FreeCAD (i.e., the initial three-dimensional geometric model is constructed), then tetrahedral mesh elements can be generated using Gmsh (i.e., mesh-variable cross-section continuous casting billet model), and then the mesh data can be converted into OpenFOAM format to complete the preprocessing task of numerical calculation.

[0060] It should be noted that, due to reasons such as technical confidentiality, complete geometric parameters of the funnel-shaped crystallizer are difficult to find in publicly available materials, and the mesh generation process is generally not described in detail. LANCE HIBBELER used ABAQUS to perform thermo-coupled three-dimensional calculations on the copper plate of the funnel-shaped crystallizer, employing experimental data from a cooperating steel mill. To compare several crystallizer structures, his paper specifically discusses the funnel surface and provides relatively detailed dimensions. This invention uses a funnel-shaped crystallizer as an example to establish a geometric model, where the width and length can be set to 1200 and 1100 mm respectively, and the maximum thickness of the upper slab can be set to 150 mm. The lower slab section of the crystallizer has a rectangular cross-section with dimensions of 1200 × 90 mm.

[0061] In the actual geometric modeling process, to improve computational efficiency, the 1 / 4 of the cast billet in the symmetrical funnel-shaped crystallizer can be used as the research object (see attached structure). Figure 2 In a funnel-shaped crystallizer, the intersection of the outer contour and the plane of symmetry on the central symmetry plane is formed by splicing together several smooth curves (such as circular arcs, spline curves, etc.). The funnel-shaped surface is a NURBS (non-uniform rational B-spline) surface bounded by the intersection lines of the aforementioned sections and straight lines. In FreeCAD, users can first draw the outer contour curves and surrounding straight lines on the symmetry plane, and then use FreeCAD's surface generation tool to automatically generate the NURBS surface. After inputting the NURBS surface, these NURBS surfaces and planes can be used to construct the initial three-dimensional geometric model of the billet in the funnel-shaped crystallizer.

[0062] S120: Perform mesh preprocessing on the initial three-dimensional geometric model to form a mesh-variable cross-section continuous casting billet model.

[0063] Specifically, the process of performing mesh preprocessing on the initial three-dimensional geometric model to form a mesh-variable cross-section continuous casting billet model includes:

[0064] Save the initial three-dimensional geometric model in Step format;

[0065] The initial three-dimensional geometric model of the Step format is processed by Gmsh to form the Gmsh grid-type variable cross-section continuous casting billet model.

[0066] Furthermore, the process of performing mesh preprocessing on the initial three-dimensional geometric model to form a mesh-variable cross-section continuous casting billet model also includes:

[0067] The Gmsh grid-type variable cross-section continuous casting billet model was converted into an OpenFOAM grid-type variable cross-section continuous casting billet model using the GmshToFoam tool.

[0068] It should be noted that in practical use, 3D geometric models generated by FreeCAD can be imported into Gmsh for mesh generation. Gmsh is a free 3D finite element mesh generator with built-in pre- and post-processing mechanisms. Its design goal is to provide a fast, lightweight mesh generation tool with controllable parameters and advanced visualization capabilities. Gmsh mainly revolves around four elements: geometry, mesh, solver, and post-processing. These controllable parameters can be input through an interactive graphical interface or via ASCII text files, implemented using Gmsh's own scripting language.

[0069] Specifically, in FreeCAD, the created 3D geometric model is saved in Step format. Step format files store the geometric information of points, lines, and surfaces of the entity. After Gmsh reads this format geometric model, it can automatically generate tetrahedral mesh elements (i.e., the Gmsh mesh-based variable cross-section continuous casting billet model). Users can control the mesh density as needed. For ease of reading, Figure 3 A model of a cast billet in a funnel-shaped crystallizer with a coarse grid is shown.

[0070] Furthermore, it should be noted that mesh files generated by Gmsh have the .msh extension, and .msh files cannot be directly used in OpenFOAM. However, OpenFOAM provides a conversion tool, gmshToFoam. Executing the gmshToFoam command within the OpenFOAM environment will convert the mesh file generated by Gmsh into the mesh file required by OpenFOAM (i.e., the OpenFOAM mesh-type variable cross-section continuous casting billet model). Additionally, mm is frequently used as the unit for drawing, while the standard SI unit m is used for OpenFOAM calculations. Therefore, within the OpenFOAM environment, it is necessary to execute the TransformPoints command, combined with the -scale command option, to convert mm to m.

[0071] S130: Determine the velocity distribution of the fluid in the grid-type variable cross-section continuous casting billet model based on a preset fluid dynamics analysis tool.

[0072] It should be noted that in a funnel-shaped molded crystallizer, the solidified billet shell moves along the pulling direction due to the pulling force, and is also deformed in cross-section due to the constraint of the mold wall. Complex turbulent flow exists within the billet shell, and complex heat transfer occurs within the continuously cast billet due to material movement. To simplify the model, the turbulent movement of the molten steel is ignored, and its heat transfer effect is accounted for by amplifying the thermal conductivity of the liquid. The funnel-shaped molded crystallizer is considered as a pipe enclosed by curved and flat surfaces. Molten steel flows in at a uniform velocity from the meniscus and flows out from the mold outlet. Due to the change in the cross-section of the molded crystallizer, the flow rate varies at different locations in the pipe, resulting in macroscopic differences in heat transfer compared to a traditional slab (with a constant channel cross-section). Therefore, in actual calculations, to calculate the heat transfer and solidification of the billet in the funnel-shaped molded crystallizer, it is necessary to first calculate the velocity distribution of the fluid in the funnel-shaped molded crystallizer as a viscous fluid, and then calculate the heat transfer and solidification of the billet in the funnel-shaped molded crystallizer.

[0073] Specifically, the incompressible fluid calculation program icoFOAM in OpenFOAM can be used to calculate the velocity distribution of the fluid in the OpenFOAM grid-type variable cross-section continuous casting billet model. The molten steel is considered a viscous medium with no friction between it and the mold wall. The meniscus is the fluid inlet, and the inlet condition is uniform velocity, determined based on the production casting speed. To eliminate the influence of the outlet boundary conditions on the fluid motion in the funnel-shaped mold, the computational domain is expanded to a sufficiently long secondary cooling zone, and the outlet condition is zero pressure gradient.

[0074] It should be noted that the momentum balance equations and their solution methods in icoFOAM are described in detail in theoretical books on fluid mechanics and heat transfer, as well as in the OpenFOAM user manual, and will not be repeated here.

[0075] S140: Based on the velocity distribution of the fluid in the grid-type variable cross-section continuous casting billet model, the temperature field distribution and solid fraction distribution of the continuous casting billet in the variable cross-section crystallizer are determined by the fluid dynamics analysis tool.

[0076] Specifically, determining the temperature field distribution and solid fraction distribution of the continuous casting billet in the variable cross-section mold using the fluid dynamics analysis tool based on the velocity distribution of the fluid in the grid-type variable cross-section continuous casting billet model includes:

[0077] Based on the fluid velocity distribution in the grid-type variable cross-section continuous casting billet model, the temperature field distribution and the solid fraction distribution are calculated using the scalarTransportFoam solver in OpenFOAM. During the calculation, the solidification heat transfer control equation for the grid-type variable cross-section continuous casting billet model is:

[0078]

[0079] In the above formula, κ eff It is the equivalent thermal conductivity, where T is temperature, ρ is density, L is latent heat of solidification, and g is the density. l C is the liquid phase fraction. p For specific heat, The heat is generated due to the movement of the billet and changes in the liquid phase fraction. This refers to the release of latent heat caused by the change in liquid phase fraction over time.

[0080] Furthermore, the process of calculating the temperature field distribution and the solid fraction distribution using the OpenFOAM scalarTransportFoam solver also includes:

[0081] An inertial term (the derivative with respect to time) is added to the solidification heat transfer steady-state control equation for the variable cross-section continuously cast billet, and the source term linearization is performed on the solidification heat transfer control equation for the variable cross-section continuously cast billet; wherein, the calculation formula for the source term linearization is as follows: in, T s and T l These are the solidus and liquidus temperatures, T 0 Δt and Δt represent the temperature at the previous moment and the calculation time step, respectively.

[0082] It should be noted that OpenFOAM provides the `scalarTransportFoam` program, which can be used to calculate scalar transport, such as heat transport, given the fluid velocity distribution in the funnel-shaped crystallizer model. However, since `scalarTransportFoam` does not include handling latent heat of solidification, additional code needs to be added to handle this. Furthermore, `scalarTransportFoam` cannot solve differential equations with variable coefficients. During continuous casting, molten steel cools in the funnel-shaped crystallizer, changing from a liquid to a solid state, undergoing a phase transition and releasing latent heat of solidification. The temperature of the cast billet also changes by several hundred degrees Celsius, resulting in significant changes in the material's thermal properties. This leads to a strongly nonlinear characteristic in the heat transfer control differential equations, making convergence difficult. On the other hand, due to the high casting speed of thin slabs and the intense cooling within the crystallizer, the temperature of the cast billet changes drastically within the crystallizer, further increasing the difficulty of solving the problem. Therefore, the important task here is to deal with the physical property parameters that change with temperature and the release of latent heat of solidification. This invention, through secondary development of OpenFOAM, can solve the heat transfer and solidification calculation of the billet in the funnel-shaped crystallizer.

[0083] Specifically, the scalarTransportFoam solver in OpenFOAM is used to solve scalar transport. For a given time-varying velocity field U, the governing equations for scalar (e.g., temperature T) transport are as follows:

[0084]

[0085] In the formula, D T This is the thermal diffusivity divided by the fluid density, assuming both the thermal diffusivity and the fluid density are constants. OpenFOAM provides a benchmark program and teaching examples for solving the above scalar transport equations. Extending from scalarTransportFoam, consider the following governing equations for heat transfer during solidification of continuously cast billets:

[0086]

[0087] 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 Liquid phase fraction. C 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. (2) The right-hand side of equation (2) is the heat source term, the first term, The heat is generated due to the movement of the billet and changes in the liquid phase fraction. (Second item from the right) This represents the release of latent heat due to the change in liquid fraction over time. Under steady-state conditions, the time derivative in equation (2) is 0, and the heat transfer equation can be simplified as follows:

[0088]

[0089] However, directly solving equation (3) is very prone to divergence. To improve the stability of the numerical solution, the following two measures are adopted.

[0090] First, we add an inertia term and still use equation (2) to solve it. Usually, we assume that the initial temperature field is uniform, the temperature is taken as the casting temperature, and the liquid phase ratio is 1. This initial field differs greatly from the calculated result of reaching the steady state, and directly solving equation (3) is very difficult to converge. Instead, we solve equation (2). The time derivative term in equation (2) is equivalent to inertia, which makes the temperature field less easy to change. Each iteration can only change the temperature field based on the previous moment, and eventually a steady state is reached over time.

[0091] Second, the source term is linearized. It appears as a source term in equation (2). In order to perform the calculation, the liquid fraction g must be known. l The variation of g with temperature T, assuming g l It has a linear relationship with T:

[0092]

[0093] In the formula T s and T l These are the solidus and liquidus temperatures, respectively. Substituting equation (4) into equation (2), the source term can be written as:

[0094]

[0095]

[0096]

[0097] In the above formula, T 0 Δt and Δt represent the temperature and calculation time step of the previous moment, respectively. A linear processing method was used in the discretization of the latent heat source term, and the proportionality coefficient in equation (5) is negative. In this way, it can be ensured that the diagonal elements dominate in the coefficient matrix of the discrete equation system, satisfying the positive definite condition of the equation.

[0098] It's worth noting that OpenFOAM uses a C++ object-oriented approach (emphasizing classes and templates, and overloading operators such as addition, subtraction, multiplication, and division) to encapsulate the field variables (scalars, vectors, and tensors) and differential algorithms in the differential equations, making the programming language very close to natural mathematical language. Therefore, the code for solving heat transfer and solidification in continuously cast billets in OpenFOAM is very concise.

[0099] Furthermore, it should be noted that OpenFOAM provides an open platform that allows variables to be declared according to the OpenFOAM format and the governing equations to be solved to be defined independently. This is the core technology of OpenFOAM secondary development, and handling physical property parameters that change with temperature is the focus of secondary development work. Taking the handling of thermal conductivity as an example, the following is an explanation with reference to the source code. In the OpenFOAM initialization file CreateFields.H, a solid-state thermal conductivity dictionary td_kappaSol is defined, and two field variables T_kappaSol and V_kappaSol are declared to record the temperature and thermal conductivity values ​​on the physical property parameter curves, respectively. This dictionary specifies that the (thermal conductivity) data table is stored in the td_kappaSol file. After executing the CreateFields.H code, the data in the file is imported into the variables T_kappaSol and V_kappaSol. In addition, the CreateFields.H file also defines the solid-state thermal conductivity kappa_Sol field variable to store the solid-state thermal conductivity; and defines the effective thermal conductivity kappa to solve the heat transfer equations described above.

[0100] Furthermore, the calculations primarily involve the third type of boundary conditions, corresponding to the mixed type of OpenFOAM, which are handled using groovyBC in this paper. groovyBC is a module in Swak4FOAM (Swiss army knife for FOAM) specifically designed for handling boundary conditions, and will not be elaborated upon here.

[0101] After calculating the temperature distribution and solidification state of the continuously cast billet in the funnel-shaped mold, the temperature distribution and solid fraction distribution at the outlet of the funnel-shaped mold can be mapped onto a two-dimensional cross-sectional grid using the mapFields tool provided by OpenFOAM; subsequently, secondary cooling calculations can be performed.

[0102] Corresponding to the above method, the present invention also provides a system for calculating the temperature distribution and solids content distribution of a variable cross-section continuously cast billet, comprising:

[0103] The modeling unit is used to construct the initial three-dimensional geometric model of the continuously cast billet in the variable cross-section crystallizer using a preset model building tool.

[0104] The preprocessing unit is used to perform mesh preprocessing on the initial three-dimensional geometric model to form a mesh-variable cross-section continuous casting billet model;

[0105] The velocity distribution calculation unit is used to determine the velocity distribution of the fluid in the grid-variable cross-section continuous casting billet model based on a preset fluid dynamics analysis tool.

[0106] The final calculation unit is used to determine the temperature field distribution and solid fraction distribution of the continuous casting billet in the variable cross-section mold based on the velocity distribution of the fluid in the grid-type variable cross-section continuous casting billet model, using the fluid dynamics analysis tool.

[0107] Furthermore, to improve the accuracy of preprocessing, the preprocessing unit provided by the present invention may include:

[0108] The first processing unit is used to save the initial three-dimensional geometric model in Step format and process the initial three-dimensional geometric model in Step format using Gmsh to form a Gmsh grid-type variable cross-section continuous casting billet model.

[0109] The second processing unit is used to convert the Gmsh grid-type variable cross-section continuous casting billet model into the OpenFOAM grid-type variable cross-section continuous casting billet model using the GmshToFoam tool.

[0110] Specifically, the final calculation unit is used to calculate the temperature field distribution and solid fraction distribution of the continuously cast billet in the variable cross-section crystallizer using the solidification heat transfer control equation; and the final calculation unit further includes:

[0111] An inertial term addition unit is used to add inertial terms to the solidification heat transfer control equation:

[0112] The source term linearization processing unit is used to perform source term linearization processing on the solidification heat transfer control equation.

[0113] In addition, to further illustrate the working principle of the method and system for calculating the temperature distribution and solid fraction distribution of the variable cross-section continuous casting billet provided by the present invention, the following examples will be used to further explain it.

[0114] To achieve proper heat transfer and solidification of the billet in the funnel-shaped mold, the flow calculations using IcoFoam assume a zero pressure gradient and a zero velocity gradient at the outlet of the computational domain. The inlet conditions are uniform velocity, determined by the production casting speed, and set to 4.5 m / min; and constant pressure, taken as atmospheric pressure. To meet the requirements of the outlet boundary conditions, the computational domain is expanded to the secondary cooling zone, extending the model from 1100 mm to 2100 mm. Otherwise, if the computational domain is not extended and the mold outlet is still used as the boundary, backflow will occur due to unsuitable boundary conditions. The extended geometry is meshed using Gmsh, forming 708,374 tetrahedral elements and 137,763 nodes (i.e., forming the Gmsh mesh-type variable cross-section continuous casting billet model). The Gmsh mesh-type variable cross-section continuous casting billet model is then converted to an OpenFOAM mesh-type variable cross-section continuous casting billet model using the GmshToFoam tool, which can then be used for flow field calculations in IcoFoam.

[0115] The magnitude of the velocity vector along the center line of the billet and its variation curve along the casting direction are calculated, as follows: Figure 4 As shown, the velocity increases with decreasing cross-sectional area, gradually leveling off at the crystallizer outlet, reaching approximately 6.0 m / min. Using ultra-low carbon (0.045%) steel and a casting temperature of 1548℃, the heat transfer and solidification of the cast billet are calculated. The liquidus temperature is taken as 1531℃, and the solidus temperature as 1510℃. The average heat exchange coefficient of the crystallizer is taken as 3000 W / (m²). 2 (℃). To save time during temperature field calculations, the extended computational domain was removed, and only the cast billet in the funnel-shaped crystallizer was considered, resulting in a newly defined computational mesh. The mapFields tool was used to map the previously calculated flow field onto the new mesh as known conditions. Heat transfer calculations were performed using a program developed based on scalarTransportFoam. The temperature field and liquidus distribution of the continuously cast billet (1100 mm from the meniscus) at the crystallizer outlet were obtained, as shown below. Figures 5-6 As shown, the calculation results are reasonable.

[0116] As can be seen from the above, the method and system for calculating the temperature distribution and solids fraction distribution of variable cross-section continuously cast billets provided by this invention establishes a simplified calculation method for billet solidification in a variable cross-section crystallizer using open-source software FreeCAD, Gmsh, and OpenFOAM, providing an economical calculation tool for secondary cooling calculations in thin slab continuous casting. Furthermore, based on scalarTransportFoam, a three-dimensional heat transfer and solidification calculation program was developed, and a calculation method to accelerate convergence was proposed for three-dimensional nonlinear heat transfer equations involving the release of latent heat of solidification. In addition, the existing OpenFOAM standard program can only be used to solve differential equations with constant coefficients and lacks processing for variable physical property parameters. This invention addresses the temperature-dependent changes in physical property parameters and non-uniform heat transfer boundaries, meeting the simulation requirements of the solidification process of continuously cast billets. Finally, by using the method and system for calculating the temperature distribution and solids fraction distribution of variable cross-section continuously cast billets provided by this invention to calculate the temperature field of variable cross-section continuously cast billets, the calculation efficiency and accuracy can be significantly improved.

[0117] The method and system for calculating the temperature distribution and solids fraction distribution of a variable cross-section continuously cast billet according to the present invention have 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 method and system for calculating the temperature distribution and solids fraction distribution of a variable cross-section continuously cast billet proposed in the present invention without departing from the scope of the 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 calculating the temperature distribution and solids fraction distribution of a variable cross-section continuously cast billet, characterized in that, include: The initial three-dimensional geometric model of the continuously cast billet in the variable cross-section crystallizer is constructed using a preset model construction tool. This includes: drawing the outer contour curve and the surrounding straight lines on the symmetry plane of the continuously cast billet in the variable cross-section crystallizer using the preset model construction tool, and generating a NURBS surface using the surface generation tool in the preset model construction tool; and constructing the initial three-dimensional geometric model based on the outer contour curve and the surrounding straight lines on the symmetry plane of the continuously cast billet in the variable cross-section crystallizer and the NURBS surface. The initial three-dimensional geometric model is preprocessed by meshing to form a mesh-variable cross-section continuous casting billet model; The velocity distribution of the fluid in the grid-type variable cross-section continuous casting billet model is determined based on a preset fluid dynamics analysis tool; wherein the fluid is considered to be a viscous fluid. Based on the fluid velocity distribution in the grid-type variable cross-section continuous casting billet model, the temperature field distribution and solid fraction distribution of the continuous casting billet in the variable cross-section crystallizer are determined using the fluid dynamics analysis tool. During the calculation, the solidification heat transfer control equation of the grid-type variable cross-section continuous casting billet model is: ； In the above formula, U is the equivalent thermal conductivity, U is the velocity field changing with time, T is the temperature, ρ is the density, and L is the latent heat of solidification. For liquid phase ratio, For specific heat, The heat is generated due to the movement of the billet and changes in the liquid phase fraction. This refers to the release of latent heat caused by the change in liquid phase fraction over time.

2. The method for calculating the temperature distribution and solids fraction distribution of a variable cross-section continuously cast billet as described in claim 1, characterized in that, The preset model building tool is FreeCAD.

3. The method for calculating the temperature distribution and solids fraction distribution of a variable cross-section continuously cast billet as described in claim 2, characterized in that, The process of performing mesh preprocessing on the initial three-dimensional geometric model to form a mesh-variable cross-section continuous casting billet model includes: Save the initial three-dimensional geometric model in Step format; The initial three-dimensional geometric model of the Step format is processed by Gmsh to form the Gmsh grid-type variable cross-section continuous casting billet model.

4. The method for calculating the temperature distribution and solids content distribution of a variable cross-section continuously cast billet as described in claim 3, characterized in that, The process of performing mesh preprocessing on the initial three-dimensional geometric model to form a mesh-variable cross-section continuous casting billet model further includes: The Gmsh grid-type variable cross-section continuous casting billet model was converted into an OpenFOAM grid-type variable cross-section continuous casting billet model using the GmshToFoam tool.

5. The method for calculating the temperature distribution and solids fraction distribution of a variable cross-section continuously cast billet as described in claim 4, characterized in that, The preset fluid dynamics analysis tool is OpenFOAM; and the process of determining the velocity distribution of the fluid in the grid-variable cross-section continuous casting billet model based on the preset fluid dynamics analysis tool includes: The velocity distribution of the fluid in the OpenFOAM grid-variable cross-section continuous casting billet model is calculated using the incompressible fluid calculation program icoFOAM in OpenFOAM.

6. The method for calculating the temperature distribution and solids fraction distribution of a variable cross-section continuously cast billet as described in claim 5, characterized in that, The determination of the temperature field distribution and solid fraction distribution of the continuous casting billet in the variable cross-section mold based on the velocity distribution of the fluid in the grid-type variable cross-section continuous casting billet model, using the fluid dynamics analysis tool, includes: Based on the velocity distribution of the fluid in the grid-type variable cross-section continuous casting billet model, the temperature field distribution and the solid fraction distribution are calculated using the scalarTransportFoam solver in OpenFOAM.

7. The method for calculating the temperature distribution and solids fraction distribution of a variable cross-section continuously cast billet as described in claim 6, characterized in that, The process of calculating the temperature field distribution and the solid fraction distribution using the OpenFOAM scalarTransportFoam solver also includes: A time derivative term is added to the solidification heat transfer control equation, and the source term is linearized. The calculation formula for the source term linearization is as follows: ;in, , , and These are the solidus and liquidus temperatures, respectively. and These represent the temperature at the previous moment and the calculation time step, respectively.

8. A system for calculating the temperature distribution and solids fraction distribution of a variable cross-section continuously cast billet, characterized in that, include: The modeling unit is used to construct an initial three-dimensional geometric model of the continuously cast billet in the variable cross-section crystallizer using a preset model construction tool; including: drawing the outer contour curve and the surrounding straight lines on the symmetry plane of the continuously cast billet in the variable cross-section crystallizer using the preset model construction tool, and generating a NURBS surface using the surface generation tool in the preset model construction tool; constructing the initial three-dimensional geometric model based on the outer contour curve and the surrounding straight lines on the symmetry plane of the continuously cast billet in the variable cross-section crystallizer and the NURBS surface; The preprocessing unit is used to perform mesh preprocessing on the initial three-dimensional geometric model to form a mesh-variable cross-section continuous casting billet model; A velocity distribution calculation unit is used to determine the velocity distribution of the fluid in the grid-type variable cross-section continuous casting billet model based on a preset fluid dynamics analysis tool; wherein the fluid is considered as a viscous fluid; The final calculation unit is used to determine the temperature field distribution and solid fraction distribution of the continuously cast billet in the variable cross-section mold, based on the velocity distribution of the fluid in the grid-type variable cross-section continuous casting billet model, using the fluid dynamics analysis tool. During the calculation, the solidification heat transfer control equation of the grid-type variable cross-section continuous casting billet model is: ； In the above formula, U is the equivalent thermal conductivity, U is the velocity field changing with time, T is the temperature, ρ is the density, and L is the latent heat of solidification. For liquid phase ratio, For specific heat, The heat is generated due to the movement of the billet and changes in the liquid phase fraction. This refers to the release of latent heat caused by the change in liquid phase fraction over time.

9. The system for calculating the temperature distribution and solids content distribution of a variable cross-section continuously cast billet as described in claim 8, characterized in that, The preprocessing unit includes: The first processing unit is used to save the initial three-dimensional geometric model in Step format and process the initial three-dimensional geometric model in Step format using Gmsh to form a Gmsh grid-type variable cross-section continuous casting billet model. The second processing unit is used to convert the Gmsh grid-type variable cross-section continuous casting billet model into the OpenFOAM grid-type variable cross-section continuous casting billet model using the GmshToFoam tool.

10. The system for calculating the temperature distribution and solids content distribution of a variable cross-section continuously cast billet as described in claim 9, characterized in that, The final calculation unit is used to calculate the temperature field distribution and solid fraction distribution of the continuously cast billet in the variable cross-section crystallizer using the solidification heat transfer control equation; and the final calculation unit further includes: An inertial term addition unit is used to add inertial terms to the solidification heat transfer control equation: The source term linearization processing unit is used to perform source term linearization processing on the solidification heat transfer control equation.

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